Optimal. Leaf size=89 \[ -\frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (-1+2 n);\frac {1}{4} (3+2 n);\sin ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-2 n) (a \sin (e+f x))^{3/2}} \]
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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 \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-1);\frac {1}{4} (2 n+3);\sin ^2(e+f x)\right )}{b f (1-2 n) (a \sin (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int \frac {(b \tan (e+f x))^n}{(a \sin (e+f x))^{3/2}} \, 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} (-1+2 n);\frac {1}{4} (3+2 n);\sin ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-2 n) (a \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 12.13, size = 90, normalized size = 1.01 \begin {gather*} \frac {2 b \cos ^2(e+f x)^{\frac {1}{2} (-1+n)} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (-1+2 n);\frac {1}{4} (3+2 n);\sin ^2(e+f x)\right ) \sqrt {a \sin (e+f x)} (b \tan (e+f x))^{-1+n}}{a^2 f (-1+2 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (b \tan \left (f x +e \right )\right )^{n}}{\left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
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 \frac {\left (b \tan {\left (e + f x \right )}\right )^{n}}{\left (a \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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 \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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