Optimal. Leaf size=48 \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]
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Rubi [A] time = 0.0270866, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2639} \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{a \csc ^3(x)} \, dx &=\frac{\sqrt{a \csc ^3(x)} \int (-\csc (x))^{3/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\frac{\sqrt{a \csc ^3(x)} \int \frac{1}{\sqrt{-\csc (x)}} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\left (\sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)\right ) \int \sqrt{\sin (x)} \, dx\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)+2 \sqrt{a \csc ^3(x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)\\ \end{align*}
Mathematica [A] time = 0.0410909, size = 46, normalized size = 0.96 \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.278, size = 343, normalized size = 7.2 \begin{align*}{\frac{\sqrt{8}\sin \left ( x \right ) }{4} \left ( 2\,\cos \left ( x \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 ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) -\cos \left ( x \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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +2\,\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 ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) -2 \right ) \sqrt{-2\,{\frac{a}{\sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) }}}} \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 \sqrt{a \csc \left (x\right )^{3}}\,{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 (\sqrt{a \csc \left (x\right )^{3}}, 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 \sqrt{a \csc ^{3}{\left (x \right )}}\, 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 \sqrt{a \csc \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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