Optimal. Leaf size=69 \[ \frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2684, 3557,
371} \begin {gather*} \frac {(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2684
Rule 3557
Rubi steps
\begin {align*} \int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \int (b \tan (e+f x))^{-m+n} \, dx\\ &=\frac {\left (b (a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \text {Subst}\left (\int \frac {x^{-m+n}}{b^2+x^2} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 67, normalized size = 0.97 \begin {gather*} \frac {a (a \cot (e+f x))^{-1+m} \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^n}{f (1-m+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (a \cot \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx\]
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 (a \cot {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, 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 (a\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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