Optimal. Leaf size=29 \[ \frac{4 \sin (x)}{3 \sqrt{\sin (2 x)}}-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0476002, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4308, 4303, 4292} \[ \frac{4 \sin (x)}{3 \sqrt{\sin (2 x)}}-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4308
Rule 4303
Rule 4292
Rubi steps
\begin{align*} \int \frac{\csc (x)}{\sin ^{\frac{3}{2}}(2 x)} \, dx &=2 \int \frac{\cos (x)}{\sin ^{\frac{5}{2}}(2 x)} \, dx\\ &=-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)}+\frac{4}{3} \int \frac{\sin (x)}{\sin ^{\frac{3}{2}}(2 x)} \, dx\\ &=-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)}+\frac{4 \sin (x)}{3 \sqrt{\sin (2 x)}}\\ \end{align*}
Mathematica [A] time = 0.0300172, size = 24, normalized size = 0.83 \[ \sqrt{\sin (2 x)} \left (\frac{\sec (x)}{2}-\frac{1}{6} \cot (x) \csc (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.039, size = 121, normalized size = 4.2 \begin{align*} -{\frac{1}{12}\sqrt{-{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \left ( 2\,\sqrt{1+\tan \left ( x/2 \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ( x/2 \right ) },1/2\,\sqrt{2} \right ) \tan \left ( x/2 \right ) - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}+1 \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \right ) }}}} \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}{\sin \left (2 \, x\right )^{\frac{3}{2}} \sin \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.88893, size = 130, normalized size = 4.48 \begin{align*} \frac{4 \, \cos \left (x\right )^{3} + \sqrt{2}{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{\cos \left (x\right ) \sin \left (x\right )} - 4 \, \cos \left (x\right )}{6 \,{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{\sin \left (2 \, x\right )^{\frac{3}{2}} \sin \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]