Optimal. Leaf size=69 \[ -\frac {\cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}-\frac {\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
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Rubi [A] time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3659, 2591, 14} \[ -\frac {\cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}-\frac {\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2591
Rule 3659
Rubi steps
\begin {align*} \int \csc ^4(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 ^4(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^{-4+n p} \left (c^2+x^2\right ) \, 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^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 {\cot ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3-n p)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 59, normalized size = 0.86 \[ \frac {\cot (e+f x) \csc ^2(e+f x) (\cos (2 (e+f x))+n p-2) \left (b (c \tan (e+f x))^n\right )^p}{f (n p-3) (n p-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 104, normalized size = 1.51 \[ \frac {{\left (2 \, \cos \left (f x + e\right )^{3} + {\left (n p - 3\right )} \cos \left (f x + e\right )\right )} e^{\left (n p \log \left (\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \relax (b)\right )}}{{\left (f n^{2} p^{2} - 4 \, f n p - {\left (f n^{2} p^{2} - 4 \, f n p + 3 \, f\right )} \cos \left (f x + e\right )^{2} + 3 \, f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{4}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 73, normalized size = 1.06 \[ \frac {\frac {b^{p} c^{n p} {\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 1\right )} \tan \left (f x + e\right )} + \frac {b^{p} c^{n p} {\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 3\right )} \tan \left (f x + e\right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\sin \left (e+f\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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