Optimal. Leaf size=82 \[ -\frac {\sin (e+f x) (a \cos (e+f x))^{m+1} (b \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\cos ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {20, 2643} \[ -\frac {\sin (e+f x) (a \cos (e+f x))^{m+1} (b \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\cos ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx &=\left ((a \cos (e+f x))^{-n} (b \cos (e+f x))^n\right ) \int (a \cos (e+f x))^{m+n} \, dx\\ &=-\frac {(a \cos (e+f x))^{1+m} (b \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+m+n);\frac {1}{2} (3+m+n);\cos ^2(e+f x)\right ) \sin (e+f x)}{a f (1+m+n) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 0.94 \[ -\frac {\sqrt {\sin ^2(e+f x)} \cot (e+f x) (a \cos (e+f x))^m (b \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\cos ^2(e+f x)\right )}{f (m+n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \cos \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,\cos \left (e+f\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos {\left (e + f x \right )}\right )^{m} \left (b \cos {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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