Optimal. Leaf size=75 \[ \frac {2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {2 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{3 b}+\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \csc ^2(a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4300, 2635, 2639} \[ \frac {2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {2 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{3 b}+\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \csc ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 2639
Rule 4300
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx &=\frac {\csc ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{3 b}+\frac {10}{3} \int \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx\\ &=-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{3 b}+2 \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 34, normalized size = 0.45 \[ \frac {2 \left (\sin ^{\frac {3}{2}}(2 (a+b x))+3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \csc \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \]
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 [A] time = 22.39, size = 137, normalized size = 1.83 \[ \frac {2 \sqrt {2}\, \left (\frac {\sqrt {2}\, \left (\sin ^{\frac {3}{2}}\left (2 b x +2 a \right )\right )}{6}-\frac {\sqrt {2}\, \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \left (2 \EllipticE \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )\right )}{4 \cos \left (2 b x +2 a \right ) \sqrt {\sin \left (2 b x +2 a \right )}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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]
________________________________________________________________________________________