Optimal. Leaf size=99 \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (-m+2 p+1)} \, _2F_1\left (\frac {1}{2} (2 p+1),\frac {1}{2} (-m+2 p+1);\frac {1}{2} (2 p+3);\sin ^2(e+f x)\right )}{f (2 p+1)} \]
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Rubi [A] time = 0.14, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3658, 2603, 2617} \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (-m+2 p+1)} \, _2F_1\left (\frac {1}{2} (2 p+1),\frac {1}{2} (-m+2 p+1);\frac {1}{2} (2 p+3);\sin ^2(e+f x)\right )}{f (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 2603
Rule 2617
Rule 3658
Rubi steps
\begin {align*} \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx &=\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \cos (e+f x))^m \tan ^{2 p}(e+f x) \, dx\\ &=\left ((d \cos (e+f x))^m \left (\frac {\sec (e+f x)}{d}\right )^m \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \left (\frac {\sec (e+f x)}{d}\right )^{-m} \tan ^{2 p}(e+f x) \, dx\\ &=\frac {(d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+2 p)} \, _2F_1\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1-m+2 p);\frac {1}{2} (3+2 p);\sin ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 81, normalized size = 0.82 \[ \frac {\tan (e+f x) \sec ^2(e+f x)^{m/2} \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \, _2F_1\left (\frac {m}{2}+1,p+\frac {1}{2};p+\frac {3}{2};-\tan ^2(e+f x)\right )}{f (2 p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.90, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x +e \right )\right )^{m} \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]
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 \[ \int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cos {\left (e + f x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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