Optimal. Leaf size=104 \[ -\frac{\cot ^5(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (5-n p)}-\frac{2 \cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}-\frac{\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
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Rubi [A] time = 0.127677, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3659, 2591, 270} \[ -\frac{\cot ^5(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (5-n p)}-\frac{2 \cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}-\frac{\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 2591
Rule 270
Rubi steps
\begin{align*} \int \csc ^6(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 \csc ^6(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac{\left (c (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int x^{-6+n p} \left (c^2+x^2\right )^2 \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac{\left (c (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int \left (c^4 x^{-6+n p}+2 c^2 x^{-4+n p}+x^{-2+n p}\right ) \, dx,x,c \tan (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)}-\frac{2 \cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}-\frac{\cot ^5(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (5-n p)}\\ \end{align*}
Mathematica [A] time = 0.287085, size = 89, normalized size = 0.86 \[ \frac{\cot (e+f x) \csc ^4(e+f x) \left (2 (n p-3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2 p^2-6 n p+8\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p-5) (n p-3) (n p-1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 9.345, size = 171293, normalized size = 1647.1 \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 = 1.2523, size = 146, normalized size = 1.4 \begin{align*} \frac{\frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 1\right )} \tan \left (f x + e\right )} + \frac{2 \, b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 3\right )} \tan \left (f x + e\right )^{3}} + \frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 5\right )} \tan \left (f x + e\right )^{5}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13383, size = 435, normalized size = 4.18 \begin{align*} \frac{{\left (8 \, \cos \left (f x + e\right )^{5} + 4 \,{\left (n p - 5\right )} \cos \left (f x + e\right )^{3} +{\left (n^{2} p^{2} - 8 \, n p + 15\right )} \cos \left (f x + e\right )\right )} 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 )}}{{\left (f n^{3} p^{3} - 9 \, f n^{2} p^{2} +{\left (f n^{3} p^{3} - 9 \, f n^{2} p^{2} + 23 \, f n p - 15 \, f\right )} \cos \left (f x + e\right )^{4} + 23 \, f n p - 2 \,{\left (f n^{3} p^{3} - 9 \, f n^{2} p^{2} + 23 \, f n p - 15 \, f\right )} \cos \left (f x + e\right )^{2} - 15 \, f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} \csc \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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