Optimal. Leaf size=86 \[ \frac {(a \cos (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {n+3}{2};\sin ^2(e+f x)\right )}{b f (n+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 86, 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 = {2603, 2617} \[ \frac {(a \cos (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {n+3}{2};\sin ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2603
Rule 2617
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \cos (e+f x))^m \left (\frac {\sec (e+f x)}{a}\right )^m\right ) \int \left (\frac {\sec (e+f x)}{a}\right )^{-m} (b \tan (e+f x))^n \, dx\\ &=\frac {(a \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+n)} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {3+n}{2};\sin ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 81, normalized size = 0.94 \[ \frac {\tan (e+f x) \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \tan (e+f x))^n \, _2F_1\left (\frac {m+2}{2},\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{f (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \cos \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.47, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a \cos {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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