Optimal. Leaf size=93 \[ \frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}}+\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2596, 2601, 2641} \[ \frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}}+\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2596
Rule 2601
Rule 2641
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{3/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{3 b^2}\\ &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {\left (a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{3 b^2 \sqrt {a \sin (e+f x)}}\\ &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 80, normalized size = 0.86 \[ \frac {2 a \sqrt {a \sin (e+f x)} \left (\sin (e+f x) \sqrt [4]{\cos ^2(e+f x)}+F\left (\left .\frac {1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )\right )}{3 b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} a \sin \left (f x + e\right )}{b^{2} \tan \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 137, normalized size = 1.47 \[ -\frac {2 \sin \left (f x +e \right ) \left (i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+\cos \left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f \left (-1+\cos \left (f x +e \right )\right ) \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2}} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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