Optimal. Leaf size=84 \[ -\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{2 b}+\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0431275, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4302, 4305} \[ -\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{2 b}+\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4302
Rule 4305
Rubi steps
\begin{align*} \int \sin (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx &=-\frac{\cos (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}+\frac{1}{2} \int \frac{\cos (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}+\frac{\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}\right )}{4 b}-\frac{\cos (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0721811, size = 72, normalized size = 0.86 \[ \frac{-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt{\sin (2 (a+b x))} \cos (a+b x)+\log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 1.171, size = 6219390, normalized size = 74040.4 \begin{align*} \text{output too large to display} \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 \, b x + 2 \, a\right )} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.538193, size = 737, normalized size = 8.77 \begin{align*} -\frac{8 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right ) - 2 \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}{\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) + 2 \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{3} -{\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{16 \, b} \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 \, b x + 2 \, a\right )} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]