Optimal. Leaf size=74 \[ \frac{\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n p+1),\frac{1}{2} (m+n p+3),-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
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Rubi [A] time = 0.0993308, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3578, 20, 3476, 364} \[ \frac{\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (1,\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
Antiderivative was successfully verified.
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Rule 3578
Rule 20
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{n p} (d \tan (e+f x))^m \, dx\\ &=\left ((c \tan (e+f x))^{-m-n p} (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{m+n p} \, dx\\ &=\frac{\left (c (c \tan (e+f x))^{-m-n p} (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int \frac{x^{m+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (1,\frac{1}{2} (1+m+n p);\frac{1}{2} (3+m+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p}{f (1+m+n p)}\\ \end{align*}
Mathematica [A] time = 0.0807976, size = 76, normalized size = 1.03 \[ \frac{\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n p+1),\frac{1}{2} (m+n p+1)+1,-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.97, size = 0, normalized size = 0. \begin{align*} \int \left ( d\tan \left ( fx+e \right ) \right ) ^{m} \left ( b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \left (c \tan{\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \tan{\left (e + f x \right )}\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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