Optimal. Leaf size=44 \[ -\frac {2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 44, 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, 2639} \[ -\frac {2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 4300
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx &=-\frac {\csc ^2(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{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 {\csc ^2(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 37, normalized size = 0.84 \[ -\frac {2 \left (E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )+\sqrt {\sin (2 (a+b x))} \cot (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \]
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 \[ \int \csc \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 17.73, size = 176, normalized size = 4.00 \[ \frac {2 \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticE \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (2 b x +2 a \right )\right )-2 \cos \left (2 b x +2 a \right )}{\cos \left (2 b x +2 a \right ) \sqrt {\sin \left (2 b x +2 a \right )}\, b} \]
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 \[ \int \csc \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
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 \[ \int \frac {\sqrt {\sin \left (2\,a+2\,b\,x\right )}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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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