Optimal. Leaf size=45 \[ \frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {4296, 2639} \[ \frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 4296
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\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 ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 41, normalized size = 0.91 \[ \frac {\sqrt {\sin (2 (a+b x))} \tan (a+b x)-E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}{\cos \left (2 \, b x + 2 \, a\right )^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 15.52, size = 83175028, normalized size = 1848333.96 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________