Optimal. Leaf size=48 \[ -\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}+\frac {2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{\sqrt {a \sin ^3(x)}} \]
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Rubi [A]
time = 0.01, 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 = {3286, 2716,
2719} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 3286
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 \begin {gather*} \frac {2 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)-\sin (2 x)}{\sqrt {a \sin ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.38, size = 318, normalized size = 6.62
method | result | size |
default | \(\frac {\left (2 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \sqrt {2}\, \cos \left (x \right )-\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \left (x \right )+2 \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \sqrt {2}-\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-2\right ) \sin \left (x \right )}{\sqrt {a \left (\sin ^{3}\left (x \right )\right )}}\) | \(318\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 102, normalized size = 2.12 \begin {gather*} \frac {{\left (-i \, \sqrt {2} \cos \left (x\right )^{2} + i \, \sqrt {2}\right )} \sqrt {-i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + {\left (i \, \sqrt {2} \cos \left (x\right )^{2} - i \, \sqrt {2}\right )} \sqrt {i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + 2 \, \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \cos \left (x\right )}{a \cos \left (x\right )^{2} - a} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a \sin ^{3}{\left (x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a\,{\sin \left (x\right )}^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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