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 = 0.0254569, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3207
Rule 2636
Rule 2641
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*}
Mathematica [A] time = 0.0678059, 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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Maple [C] time = 0.283, size = 360, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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