Optimal. Leaf size=123 \[ -\frac {154 \sin (x) \cos (x)}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2639} \[ -\frac {154 \sin (x) \cos (x)}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx &=\frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \sin ^3(x)}}\\ &=-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (11 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \sin ^3(x)}}\\ &=-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \sin ^3(x)}}\\ &=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \sin ^3(x)}}\\ &=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx}{195 a^2 \sqrt {a \sin ^3(x)}}\\ &=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{195 a^2 \sqrt {a \sin ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 60, normalized size = 0.49 \[ -\frac {2 \left (231 \sin (x) \cos (x)+\cot (x) \left (45 \csc ^4(x)+55 \csc ^2(x)+77\right )-231 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{585 a^2 \sqrt {a \sin ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}}{{\left (a^{3} \cos \relax (x)^{8} - 4 \, a^{3} \cos \relax (x)^{6} + 6 \, a^{3} \cos \relax (x)^{4} - 4 \, a^{3} \cos \relax (x)^{2} + a^{3}\right )} \sin \relax (x)}, x\right ) \]
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 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 1301, normalized size = 10.58 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,{\sin \relax (x)}^3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin ^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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