Optimal. Leaf size=48 \[ \frac {2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{\sqrt {a \sin ^3(x)}}-\frac {2 \sin (x) \cos (x)}{\sqrt {a \sin ^3(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2639} \[ \frac {2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{\sqrt {a \sin ^3(x)}}-\frac {2 \sin (x) \cos (x)}{\sqrt {a \sin ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx &=\frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {3}{2}}(x)} \, dx}{\sqrt {a \sin ^3(x)}}\\ &=-\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}-\frac {\sin ^{\frac {3}{2}}(x) \int \sqrt {\sin (x)} \, dx}{\sqrt {a \sin ^3(x)}}\\ &=-\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}+\frac {2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{\sqrt {a \sin ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.77 \[ \frac {2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )-\sin (2 x)}{\sqrt {a \sin ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}}{{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \sin \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 330, normalized size = 6.88 \[ \frac {\left (2 \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \cos \relax (x ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \cos \relax (x ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-2\right ) \sin \relax (x )}{\sqrt {a \left (\sin ^{3}\relax (x )\right )}} \]
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 \[ \int \frac {1}{\sqrt {a \sin \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a\,{\sin \relax (x)}^3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \sin ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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