Optimal. Leaf size=86 \[ -\frac {\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {1}{b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 86, 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 = {2597, 2601, 2641} \[ -\frac {\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {1}{b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2597
Rule 2601
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \sin (e+f x)} (b \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {\int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{2 b^2}\\ &=-\frac {1}{b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {\left (\sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{2 b^2 \sqrt {a \sin (e+f x)}}\\ &=-\frac {1}{b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 79, normalized size = 0.92 \[ \frac {\sin (e+f x) \left (-F\left (\left .\frac {1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )\right )-\sqrt [4]{\cos ^2(e+f x)}}{b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}{a b^{2} \sin \left (f x + e\right ) \tan \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 185, normalized size = 2.15 \[ -\frac {\left (i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )\right ) \sin \left (f x +e \right )}{f \sqrt {a \sin \left (f x +e \right )}\, \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2}} \]
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 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a\,\sin \left (e+f\,x\right )}\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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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