Optimal. Leaf size=92 \[ \frac {\cos ^2(e+f x)^{\frac {1}{2}+p} \, _2F_1\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1+m+2 p);\frac {1}{2} (3+m+2 p);\sin ^2(e+f x)\right ) (d \sin (e+f x))^m \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p)} \]
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Rubi [A]
time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3739, 2682,
2657} \begin {gather*} \frac {\tan (e+f x) \cos ^2(e+f x)^{p+\frac {1}{2}} \left (b \tan ^2(e+f x)\right )^p (d \sin (e+f x))^m \, _2F_1\left (\frac {1}{2} (2 p+1),\frac {1}{2} (m+2 p+1);\frac {1}{2} (m+2 p+3);\sin ^2(e+f x)\right )}{f (m+2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rule 3739
Rubi steps
\begin {align*} \int (d \sin (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 (d \sin (e+f x))^m \tan ^{2 p}(e+f x) \, dx\\ &=\left (d \cos ^{2 p}(e+f x) \sin (e+f x) (d \sin (e+f x))^{-1-2 p} \left (b \tan ^2(e+f x)\right )^p\right ) \int \cos ^{-2 p}(e+f x) (d \sin (e+f x))^{m+2 p} \, dx\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2}+p} \, _2F_1\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1+m+2 p);\frac {1}{2} (3+m+2 p);\sin ^2(e+f x)\right ) (d \sin (e+f x))^m \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 1.37, size = 292, normalized size = 3.17 \begin {gather*} \frac {(3+m+2 p) F_1\left (\frac {1}{2}+\frac {m}{2}+p;2 p,1+m;\frac {3}{2}+\frac {m}{2}+p;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x) (d \sin (e+f x))^m \left (b \tan ^2(e+f x)\right )^p}{f (1+m+2 p) \left ((3+m+2 p) F_1\left (\frac {1}{2}+\frac {m}{2}+p;2 p,1+m;\frac {3}{2}+\frac {m}{2}+p;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((1+m) F_1\left (\frac {3}{2}+\frac {m}{2}+p;2 p,2+m;\frac {5}{2}+\frac {m}{2}+p;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 p F_1\left (\frac {3}{2}+\frac {m}{2}+p;1+2 p,1+m;\frac {5}{2}+\frac {m}{2}+p;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{m} \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 \sin {\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\,\sin \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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