Optimal. Leaf size=80 \[ \frac {(d \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \]
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Rubi [A]
time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3740, 2684,
3557, 371} \begin {gather*} \frac {\tan (e+f x) (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (1,\frac {1}{2} (-m+n p+1);\frac {1}{2} (-m+n p+3);-\tan ^2(e+f x)\right )}{f (-m+n p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2684
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int (d \cot (e+f x))^m \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 (d \cot (e+f x))^m (c \tan (e+f x))^{n p} \, dx\\ &=\left ((d \cot (e+f x))^m (c \tan (e+f x))^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{-m+n p} \, dx\\ &=\frac {\left (c (d \cot (e+f x))^m (c \tan (e+f x))^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \text {Subst}\left (\int \frac {x^{-m+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac {(d \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 77, normalized size = 0.96 \begin {gather*} \frac {d (d \cot (e+f x))^{-1+m} \, _2F_1\left (1,\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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} \left (d \cot {\left (e + f x \right )}\right )^{m}\, 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 = -0.01 \begin {gather*} \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\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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