Optimal. Leaf size=130 \[ -\frac {\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{6 a^2 b^2 f \sqrt {a \sin (e+f x)}}+\frac {1}{6 a^2 b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.17, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2597, 2599, 2601, 2641} \[ -\frac {\sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{6 a^2 b^2 f \sqrt {a \sin (e+f x)}}+\frac {1}{6 a^2 b f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2597
Rule 2599
Rule 2601
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{(a \sin (e+f x))^{5/2} (b \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {\int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx}{6 b^2}\\ &=-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}+\frac {1}{6 a^2 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}{12 a^2 b^2}\\ &=-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}+\frac {1}{6 a^2 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}{12 a^2 b^2 \sqrt {a \sin (e+f x)}}\\ &=-\frac {1}{3 b f (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}+\frac {1}{6 a^2 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)}}{6 a^2 b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 96, normalized size = 0.74 \[ \frac {\sqrt [4]{\cos ^2(e+f x)} \left (1-2 \csc ^2(e+f x)\right )-\sin (e+f x) F\left (\left .\frac {1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )}{6 a^2 b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}{{\left (a^{3} b^{2} \cos \left (f x + e\right )^{2} - a^{3} b^{2}\right )} \sin \left (f x + e\right ) \tan \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.53, size = 337, normalized size = 2.59 \[ \frac {\left (i \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+i \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right )-\cos \left (f x +e \right )\right ) \sin \left (f x +e \right )}{6 f \left (a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2}} \]
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 \[ \int \frac {1}{\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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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