Optimal. Leaf size=89 \[ \frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (5+2 n);\frac {1}{4} (9+2 n);\sin ^2(e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{1+n}}{b f (5+2 n)} \]
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Rubi [A]
time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2682, 2657}
\begin {gather*} \frac {2 (a \sin (e+f x))^{3/2} \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n+5);\frac {1}{4} (2 n+9);\sin ^2(e+f x)\right )}{b f (2 n+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx &=\frac {\left (a \cos ^{1+n}(e+f x) (a \sin (e+f x))^{-1-n} (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) (a \sin (e+f x))^{\frac {3}{2}+n} \, dx}{b}\\ &=\frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (5+2 n);\frac {1}{4} (9+2 n);\sin ^2(e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{1+n}}{b f (5+2 n)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 33.33, size = 297, normalized size = 3.34 \begin {gather*} \frac {8 (9+2 n) F_1\left (\frac {5}{4}+\frac {n}{2};n,\frac {5}{2};\frac {9}{4}+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n}{f (5+2 n) \left (2 (9+2 n) F_1\left (\frac {5}{4}+\frac {n}{2};n,\frac {5}{2};\frac {9}{4}+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (5 F_1\left (\frac {9}{4}+\frac {n}{2};n,\frac {7}{2};\frac {13}{4}+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 n F_1\left (\frac {9}{4}+\frac {n}{2};1+n,\frac {5}{2};\frac {13}{4}+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (b \tan \left (f x +e \right )\right )^{n}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\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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