Optimal. Leaf size=31 \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
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Rubi [A] time = 0.0925313, antiderivative size = 31, 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 = {3659, 2607, 32} \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(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 \sec ^2(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac{\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int (c x)^{n p} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end{align*}
Mathematica [A] time = 0.0228926, size = 31, normalized size = 1. \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.749, size = 25070, normalized size = 808.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03106, size = 47, normalized size = 1.52 \begin{align*} \frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )}{{\left (n p + 1\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42176, size = 126, normalized size = 4.06 \begin{align*} \frac{e^{\left (n p \log \left (\frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \left (b\right )\right )} \sin \left (f x + e\right )}{{\left (f n p + f\right )} \cos \left (f x + e\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} \sec ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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