Optimal. Leaf size=72 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (1+m+2 p);\frac {1}{2} (3+m+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p)} \]
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Rubi [A]
time = 0.05, antiderivative size = 72, 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, 20, 3557,
371} \begin {gather*} \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \tan (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (m+2 p+1);\frac {1}{2} (m+2 p+3);-\tan ^2(e+f x)\right )}{f (m+2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 3557
Rule 3659
Rubi steps
\begin {align*} \int (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx &=\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{2 p}(e+f x) (d \tan (e+f x))^m \, dx\\ &=\left (\tan ^{-m-2 p}(e+f x) (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{m+2 p}(e+f x) \, dx\\ &=\frac {\left (\tan ^{-m-2 p}(e+f x) (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p\right ) \text {Subst}\left (\int \frac {x^{m+2 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+m+2 p);\frac {1}{2} (3+m+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 74, normalized size = 1.03 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2} (1+m+2 p);1+\frac {1}{2} (1+m+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \left (d \tan \left (f x +e \right )\right )^{m} \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \tan {\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 {tan}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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