Optimal. Leaf size=63 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (1+m+n);\frac {1}{2} (3+m+n);-\tan ^2(e+f x)\right ) (a \tan (e+f x))^{1+m} (b \tan (e+f x))^n}{a f (1+m+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, 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 = {20, 3557, 371}
\begin {gather*} \frac {(a \tan (e+f x))^{m+1} (b \tan (e+f x))^n \, _2F_1\left (1,\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);-\tan ^2(e+f x)\right )}{a f (m+n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 3557
Rubi steps
\begin {align*} \int (a \tan (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \tan (e+f x))^{-n} (b \tan (e+f x))^n\right ) \int (a \tan (e+f x))^{m+n} \, dx\\ &=\frac {\left (a (a \tan (e+f x))^{-n} (b \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {x^{m+n}}{a^2+x^2} \, dx,x,a \tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+m+n);\frac {1}{2} (3+m+n);-\tan ^2(e+f x)\right ) (a \tan (e+f x))^{1+m} (b \tan (e+f x))^n}{a f (1+m+n)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 66, normalized size = 1.05 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2} (1+m+n);1+\frac {1}{2} (1+m+n);-\tan ^2(e+f x)\right ) \tan (e+f x) (a \tan (e+f x))^m (b \tan (e+f x))^n}{f (1+m+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \left (a \tan \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx\]
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 \tan {\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 = -0.02 \begin {gather*} \int {\left (a\,\mathrm {tan}\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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