Powers of the Same Base

Combine powers of the same base.

syms x y z
combine(x^y*x^z)

ans =
x^(y + z)

Combine powers of numeric arguments. To prevent MATLAB^® from evaluating the expression, use sym to convert at least one numeric argument into a symbolic value.

syms x y
combine(x^(3)*x^y*x^exp(sym(1)))

ans =
x^(y + exp(1) + 3)

Here, sym converts 1 into a symbolic value, preventing MATLAB from evaluating the expression e1.

Powers of the Same Exponent

Combine powers with the same exponents in certain cases.

combine(sqrt(sym(2))*sqrt(3))

ans =
6^(1/2)

combine does not usually combine the powers because the internal simplifier applies the same rules in the opposite direction to expand the result.

syms x y 
combine(y^5*x^5)

ans =
x^5*y^5

Terms with Logarithms

Combine terms with logarithms by specifying the target argument as log. For real positive numbers, the logarithm of a product equals the sum of the logarithms of its factors.

S = log(sym(2)) + log(sym(3));
combine(S,'log')

ans =
log(6)

Try combining log(a) + log(b). Because a and b are assumed to be complex numbers by default, the rule does not hold and combine does not combine the terms.

syms a b
S = log(a) + log(b);
combine(S,'log')

ans =
log(a) + log(b)

Apply the rule by setting assumptions such that a and b satisfy the conditions for the rule.

assume(a > 0)
assume(b > 0)
S = log(a) + log(b);
combine(S,'log')

ans =
log(a*b)

For future computations, clear the assumptions set on variables a and b by recreating them using syms.

syms a b

Alternatively, apply the rule by ignoring analytic constraints using 'IgnoreAnalyticConstraints'.

syms a b
S = log(a) + log(b);
combine(S,'log','IgnoreAnalyticConstraints',true)

ans =
 log(a*b)

Terms with Sine and Cosine Function Calls

Rewrite products of sine and cosine functions as a sum of the functions by setting the target argument to sincos.

syms a b
combine(sin(a)*cos(b) + sin(b)^2,'sincos')

ans =
sin(a + b)/2 - cos(2*b)/2 + sin(a - b)/2 + 1/2

Rewrite sums of sine and cosine functions by setting the target argument to sincos.

combine(cos(a) + sin(a),'sincos')

ans =
2^(1/2)*cos(a - pi/4)

Rewrite a cosine squared function by setting the target argument to sincos.

combine(cos(a)^2,'sincos')

ans =
cos(2*a)/2 + 1/2

combine does not rewrite powers of sine or cosine functions with negative integer exponents.

syms a b
combine(sin(b)^(-2)*cos(b)^(-2),'sincos')

ans =
1/(cos(b)^2*sin(b)^2)

Exponential Terms

Combine terms with exponents by specifying the target argument as exp.

combine(exp(sym(3))*exp(sym(2)),'exp')

ans =
exp(5)

syms a
combine(exp(a)^3, 'exp')

ans =
exp(3*a)

Terms with Integrals

Combine terms with integrals by specifying the target argument as int.

syms a f(x) g(x)
combine(int(f(x),x)+int(g(x),x),'int')
combine(a*int(f(x),x),'int')

ans =
int(f(x) + g(x), x)
ans =
int(a*f(x), x)

Combine integrals with the same limits.

syms a b h(z)
combine(int(f(x),x,a,b)+int(h(z),z,a,b),'int')

ans =
int(f(x) + h(x), x, a, b)

Terms with Inverse Tangent Function Calls

Combine two calls to the inverse tangent function by specifying the target argument as atan.

syms a b
assume(-1 < a < 1)
assume(-1 < b < 1)
combine(atan(a) + atan(b),'atan')

ans =
-atan((a + b)/(a*b - 1))

Combine two calls to the inverse tangent function. combine simplifies the expression to a symbolic value if possible.

assume(a > 0)
combine(atan(a) + atan(1/a),'atan')

ans =
pi/2

For further computations, clear the assumptions:

syms a b

Terms with Calls to Gamma Function

Combine multiple gamma functions by specifying the target as gamma.

syms x
combine(gamma(x)*gamma(1-x),'gamma')

ans =
 -pi/sin(pi*(x - 1))

combine simplifies quotients of gamma functions to rational expressions.

Multiple Input Expressions in One Call

Evaluate multiple expressions in one function call by using a symbolic matrix as the input parameter.

S = [sqrt(sym(2))*sqrt(5), sqrt(2)*sqrt(sym(11))];
combine(S)

ans =
[ 10^(1/2), 22^(1/2)]