Optimal. Leaf size=106 \[ \frac {6 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b}-\frac {2 \sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac {2 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b} \]
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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, 2635, 2639} \[ \frac {6 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b}-\frac {2 \sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac {2 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 4300
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx &=\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+\frac {18}{7} \int \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx\\ &=-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+2 \int \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx\\ &=-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+\frac {6}{5} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 66, normalized size = 0.62 \[ \frac {\sqrt {\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))+84 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{70 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (2 \, b x + 2 \, a\right )^{4} - 2 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 1\right )} \csc \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 29.08, size = 204, normalized size = 1.92 \[ \frac {8 \sqrt {2}\, \left (\frac {\sqrt {2}\, \left (\sin ^{\frac {7}{2}}\left (2 b x +2 a \right )\right )}{56}-\frac {\sqrt {2}\, \left (6 \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticE \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{4}\left (2 b x +2 a \right )\right )+2 \left (\sin ^{2}\left (2 b x +2 a \right )\right )\right )}{80 \cos \left (2 b x +2 a \right ) \sqrt {\sin \left (2 b x +2 a \right )}}\right )}{b} \]
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 \[ \int \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{9/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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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