Optimal. Leaf size=62 \[ -\frac {\cot ^2(e+f x) \, _2F_1\left (1,\frac {1}{2} (n p-2);\frac {n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3659, 16, 3476, 364} \[ -\frac {\cot ^2(e+f x) \, _2F_1\left (1,\frac {1}{2} (n p-2);\frac {n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \cot ^3(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 \cot ^3(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (c^3 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{-3+n p} \, dx\\ &=\frac {\left (c^4 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname {Subst}\left (\int \frac {x^{-3+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^2(e+f x) \, _2F_1\left (1,\frac {1}{2} (-2+n p);\frac {n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.95 \[ \frac {\cot ^2(e+f x) \, _2F_1\left (1,\frac {n p}{2}-1;\frac {n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p-2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{3}, x\right ) \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.36, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{3}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
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 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cot ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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