Optimal. Leaf size=50 \[ \frac {\, _2F_1\left (1,\frac {n p}{2};1+\frac {n p}{2};-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
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Rubi [A]
time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3740, 16, 3557,
371} \begin {gather*} \frac {\, _2F_1\left (1,\frac {n p}{2};\frac {n p}{2}+1;-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \cot (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 (e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (c (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{-1+n p} \, dx\\ &=\frac {\left (c^2 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \text {Subst}\left (\int \frac {x^{-1+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac {\, _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}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} \frac {\, _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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \cot \left (f x +e \right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cot {\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {cot}\left (e+f\,x\right )\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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