Optimal. Leaf size=40 \[ \frac{E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0357468, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4298, 2639} \[ \frac{E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4298
Rule 2639
Rubi steps
\begin{align*} \int \sin ^2(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx &=-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b}+\frac{1}{2} \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=\frac{E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{2 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0827766, size = 34, normalized size = 0.85 \[ -\frac{\sin ^{\frac{3}{2}}(2 (a+b x))-3 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 3.85, size = 16542587, normalized size = 413564.7 \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 )^{2}\,{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 (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}, x\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 \, b x + 2 \, a\right )} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]