Optimal. Leaf size=48 \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0173165, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2639} \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 2636
Rule 2639
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*}
Mathematica [A] time = 0.0265732, size = 37, normalized size = 0.77 \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )-\sin (2 x)}{\sqrt{a \sin ^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.257, size = 330, normalized size = 6.9 \begin{align*}{\sin \left ( x \right ) \left ( 2\,\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \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 ) }}}{\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{2}\cos \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \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 ) }}}{\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{2}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \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 ) }}}{\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{2}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \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 ) }}}{\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 ){\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right )^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}}{{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin ^{3}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right )^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]