Optimal. Leaf size=77 \[ -\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {10 \sin ^{\frac {3}{2}}(x) F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2641} \[ -\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {10 \sin ^{\frac {3}{2}}(x) F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx &=\frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \sin ^3(x)}}\\ &=-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}}+\frac {\left (5 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \sin ^3(x)}}\\ &=-\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}}+\frac {\left (5 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx}{21 a \sqrt {a \sin ^3(x)}}\\ &=-\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}}-\frac {10 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{21 a \sqrt {a \sin ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 48, normalized size = 0.62 \[ -\frac {2 \sin ^2(x) \left (3 \cot (x)+5 \sin (x) \cos (x)+5 \sin ^{\frac {5}{2}}(x) F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{21 \left (a \sin ^3(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}}{a^{2} \cos \relax (x)^{6} - 3 \, a^{2} \cos \relax (x)^{4} + 3 \, a^{2} \cos \relax (x)^{2} - a^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.46, size = 360, normalized size = 4.68 \[ -\frac {\left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (5 i \sqrt {2}\, \sin \relax (x ) \left (\cos ^{3}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+5 i \sqrt {2}\, \sin \relax (x ) \left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-5 i \sqrt {2}\, \sin \relax (x ) \cos \relax (x ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )-5 i \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sin \relax (x )-10 \left (\cos ^{3}\relax (x )\right )+16 \cos \relax (x )\right )}{21 \left (a \left (\sin ^{3}\relax (x )\right )\right )^{\frac {3}{2}} \sin \relax (x )^{3}} \]
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 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,{\sin \relax (x)}^3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin ^{3}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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