Optimal. Leaf size=48 \[ \frac {2 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b}-\frac {\sqrt {\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b} \]
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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 = {4300, 2641} \[ \frac {2 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b}-\frac {\sqrt {\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 4300
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx &=-\frac {\csc ^2(a+b x) \sqrt {\sin (2 a+2 b x)}}{3 b}+\frac {2}{3} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{3 b}-\frac {\csc ^2(a+b x) \sqrt {\sin (2 a+2 b x)}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 82, normalized size = 1.71 \[ -\frac {\sqrt {\sin (2 (a+b x))} \csc ^2(a+b x)+\frac {\sqrt {2} (\sin (a+b x)+\cos (a+b x)) F\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac {1}{2}\right )}{\sqrt {\sin (2 (a+b x))+1}}}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\csc \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}\left (b x +a \right )}{\sqrt {\sin \left (2 b x +2 a \right )}}\, dx \]
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 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,\sqrt {\sin \left (2\,a+2\,b\,x\right )}} \,d x \]
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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