Optimal. Leaf size=106 \[ \frac {30 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{77 b}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4300, 2636, 2641} \[ \frac {30 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{77 b}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 4300
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx &=-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {18}{11} \int \frac {1}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {90}{77} \int \frac {1}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {30}{77} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {30 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{77 b}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 86, normalized size = 0.81 \[ \frac {480 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )+\sqrt {\sin (2 (a+b x))} \left (-7 \csc ^6(a+b x)-32 \csc ^4(a+b x)-141 \csc ^2(a+b x)+11 \sec ^2(a+b x) \left (\sec ^2(a+b x)+9\right )\right )}{1232 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\csc \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{4} - 2 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}\left (b x +a \right )}{\sin \left (2 b x +2 a \right )^{\frac {9}{2}}}\, dx \]
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 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{9/2}} \,d x \]
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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