Optimal. Leaf size=59 \[ \frac{\tan (e+f x) \left (b \tan ^n(e+f x)\right )^p \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\tan ^2(e+f x)\right )}{f (n p+1)} \]
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Rubi [A] time = 0.0400539, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3659, 3476, 364} \[ \frac{\tan (e+f x) \left (b \tan ^n(e+f x)\right )^p \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right )}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (b \tan ^n(e+f x)\right )^p \, dx &=\left (\tan ^{-n p}(e+f x) \left (b \tan ^n(e+f x)\right )^p\right ) \int \tan ^{n p}(e+f x) \, dx\\ &=\frac{\left (\tan ^{-n p}(e+f x) \left (b \tan ^n(e+f x)\right )^p\right ) \operatorname{Subst}\left (\int \frac{x^{n p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (1,\frac{1}{2} (1+n p);\frac{1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^n(e+f x)\right )^p}{f (1+n p)}\\ \end{align*}
Mathematica [A] time = 0.0413329, size = 57, normalized size = 0.97 \[ \frac{\tan (e+f x) \left (b \tan ^n(e+f x)\right )^p \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\tan ^2(e+f x)\right )}{f n p+f} \]
Antiderivative was successfully verified.
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Maple [F] time = 10.709, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \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 (b \tan \left (f x + e\right )^{n}\right )^{p}\,{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 (b \tan \left (f x + e\right )^{n}\right )^{p}, 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 \tan ^{n}{\left (e + f x \right )}\right )^{p}\, 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 (b \tan \left (f x + e\right )^{n}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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