Optimal. Leaf size=89 \[ -\frac {2 \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\sin ^2(e+f x)\right )}{b f (1-2 n) (a \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 0.12, 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 = {2602, 2577} \[ -\frac {2 \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\sin ^2(e+f x)\right )}{b f (1-2 n) (a \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
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 = 1.96, size = 90, normalized size = 1.01 \[ \frac {2 b \sqrt {a \sin (e+f x)} \cos ^2(e+f x)^{\frac {n-1}{2}} (b \tan (e+f x))^{n-1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\sin ^2(e+f x)\right )}{a^2 f (2 n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - a^{2}}, x\right ) \]
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 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{n}}{\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, 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 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{n}}{\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (b \tan {\left (e + f x \right )}\right )^{n}}{\left (a \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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