Optimal. Leaf size=99 \[ \frac{\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+2 p+1)} \text{Hypergeometric2F1}\left (\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1),\frac{1}{2} (2 p+3),\sin ^2(e+f x)\right )}{f (2 p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13651, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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.
[In]
[Out]
Rule 3658
Rule 2603
Rule 2617
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*}
Mathematica [A] time = 0.500143, size = 81, normalized size = 0.82 \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \text{Hypergeometric2F1}\left (\frac{m}{2}+1,p+\frac{1}{2},p+\frac{3}{2},-\tan ^2(e+f x)\right )}{f (2 p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.706, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{m} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]