Optimal. Leaf size=50 \[ \frac{\text{Hypergeometric2F1}\left (1,\frac{n p}{2},\frac{n p}{2}+1,-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
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Rubi [A] time = 0.0800734, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3659, 16, 3476, 364} \[ \frac{\, _2F_1\left (1,\frac{n p}{2};\frac{n p}{2}+1;-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 16
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \cot (e+f x) \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 \cot (e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (c (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{-1+n p} \, dx\\ &=\frac{\left (c^2 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int \frac{x^{-1+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (1,\frac{n p}{2};1+\frac{n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p}\\ \end{align*}
Mathematica [A] time = 0.0158514, size = 50, normalized size = 1. \[ \frac{\text{Hypergeometric2F1}\left (1,\frac{n p}{2},\frac{n p}{2}+1,-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.779, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( fx+e \right ) \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} \cot \left (f x + e\right )\,{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} \cot \left (f x + e\right ), 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} \cot{\left (e + f x \right )}\, 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} \cot \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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