Optimal. Leaf size=33 \[ -\frac {\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
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Rubi [A] time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3659, 2591, 30} \[ -\frac {\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2591
Rule 3659
Rubi steps
\begin {align*} \int \csc ^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 \csc ^2(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^{-2+n p} \, 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)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 31, normalized size = 0.94 \[ \frac {\cot (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 51, normalized size = 1.55 \[ \frac {\cos \left (f x + e\right ) e^{\left (n p \log \left (\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \relax (b)\right )}}{{\left (f n p - f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )^{2}\,{d x} \]
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 \[ \int \left (\csc ^{2}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 37, normalized size = 1.12 \[ \frac {b^{p} c^{n p} {\left (\tan \left (f x + e\right )^{n}\right )}^{p}}{{\left (n p - 1\right )} f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\sin \left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \csc ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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