Optimal. Leaf size=97 \[ \frac{\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+n p+1)} \left (b (c \tan (e+f x))^n\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1),\frac{1}{2} (n p+3),\sin ^2(e+f x)\right )}{f (n p+1)} \]
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Rubi [A] time = 0.107514, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3659, 2617} \[ \frac{\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+n p+1)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right )}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 2617
Rubi steps
\begin{align*} \int (d \sec (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 (d \sec (e+f x))^m (c \tan (e+f x))^{n p} \, dx\\ &=\frac{\cos ^2(e+f x)^{\frac{1}{2} (1+m+n p)} \, _2F_1\left (\frac{1}{2} (1+n p),\frac{1}{2} (1+m+n p);\frac{1}{2} (3+n p);\sin ^2(e+f x)\right ) (d \sec (e+f x))^m \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end{align*}
Mathematica [A] time = 0.156979, size = 89, normalized size = 0.92 \[ \frac{\cot (e+f x) (d \sec (e+f x))^m \left (-\tan ^2(e+f x)\right )^{\frac{1}{2} (1-n p)} \left (b (c \tan (e+f x))^n\right )^p \text{Hypergeometric2F1}\left (\frac{m}{2},\frac{1}{2} (1-n p),\frac{m+2}{2},\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \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 \sec \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 \sec \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 \sec{\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 \sec \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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