Optimal. Leaf size=48 \[ \frac {\sin ^2(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{6 b} \]
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Rubi [A] time = 0.04, antiderivative size = 48, 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, 2641} \[ \frac {\sin ^2(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 4296
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^2(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {1}{6} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{6 b}+\frac {\sin ^2(a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 83, normalized size = 1.73 \[ \frac {\sqrt {\sin (2 (a+b x))} \sec ^2(a+b x)-\frac {\sqrt {2} (\sin (a+b x)+\cos (a+b x)) F\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac {1}{2}\right )}{\sqrt {\sin (2 (a+b x))+1}}}{12 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right )^{2} - 1}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 59.89, size = 123, normalized size = 2.56 \[ \frac {\sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (2 b x +2 a \right )-2 \left (\cos ^{2}\left (2 b x +2 a \right )\right )+2 \cos \left (2 b x +2 a \right )}{12 \sin \left (2 b x +2 a \right )^{\frac {3}{2}} \cos \left (2 b x +2 a \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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