Optimal. Leaf size=45 \[ -\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \sqrt{\sin (2 x)} \cos (x)+\frac{1}{4} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0297949, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4302, 4305} \[ -\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \sqrt{\sin (2 x)} \cos (x)+\frac{1}{4} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4302
Rule 4305
Rubi steps
\begin{align*} \int \sin (x) \sqrt{\sin (2 x)} \, dx &=-\frac{1}{2} \cos (x) \sqrt{\sin (2 x)}+\frac{1}{2} \int \frac{\cos (x)}{\sqrt{\sin (2 x)}} \, dx\\ &=-\frac{1}{4} \sin ^{-1}(\cos (x)-\sin (x))+\frac{1}{4} \log \left (\cos (x)+\sin (x)+\sqrt{\sin (2 x)}\right )-\frac{1}{2} \cos (x) \sqrt{\sin (2 x)}\\ \end{align*}
Mathematica [A] time = 0.0352929, size = 41, normalized size = 0.91 \[ \frac{1}{4} \left (-\sin ^{-1}(\cos (x)-\sin (x))-2 \sqrt{\sin (2 x)} \cos (x)+\log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.053, size = 171, normalized size = 3.8 \begin{align*}{\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 ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) +2\, \left ( \tan \left ( x/2 \right ) \right ) ^{3}-2\,\tan \left ( x/2 \right ) \right ){\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 ) }}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \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 \sqrt{\sin \left (2 \, x\right )} \sin \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.97212, size = 513, normalized size = 11.4 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} \cos \left (x\right ) + \frac{1}{8} \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac{1}{8} \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac{1}{16} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (x\right )^{3} -{\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt{\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\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 \sqrt{\sin \left (2 \, x\right )} \sin \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]