Optimal. Leaf size=40 \[ \frac {F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 40, 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 = {4298, 2641} \[ \frac {F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 4298
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx &=-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b}+\frac {1}{2} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 75, normalized size = 1.88 \[ -\frac {2 \sqrt {\sin (2 (a+b x))}+\frac {\sqrt {2} (\sin (a+b x)+\cos (a+b x)) F\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac {1}{2}\right )}{\sqrt {\sin (2 (a+b x))+1}}}{4 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\cos \left (b x + a\right )^{2} - 1}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]
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 [B] time = 13.51, size = 53360209, normalized size = 1334005.22 \[ \text {output too large to display} \]
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 \[ \int \frac {\sin \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \]
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 \[ \int \frac {{\sin \left (a+b\,x\right )}^2}{\sqrt {\sin \left (2\,a+2\,b\,x\right )}} \,d x \]
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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