Optimal. Leaf size=106 \[ \frac {2 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b}-\frac {2 \sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}-\frac {2 \sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x) \csc ^2(a+b x)}{5 b} \]
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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, 2641} \[ \frac {2 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b}-\frac {2 \sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}-\frac {2 \sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x) \csc ^2(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 4300
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx &=\frac {\csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{5 b}+\frac {14}{5} \int \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx\\ &=-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{5 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{5 b}+2 \int \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=-\frac {2 \cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{3 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{5 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{5 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 {2 \cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{3 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{5 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 76, normalized size = 0.72 \[ \frac {9 \sin (2 (a+b x))-10 \sin (4 (a+b x))-3 \sin (6 (a+b x))+20 \sqrt {\sin (2 (a+b x))} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{30 b \sqrt {\sin (2 (a+b x))}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}, x\right ) \]
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 = 25.25, size = 139, normalized size = 1.31 \[ \frac {4 \sqrt {2}\, \left (\frac {\sqrt {2}\, \left (\sin ^{\frac {5}{2}}\left (2 b x +2 a \right )\right )}{20}+\frac {\sqrt {2}\, \left (\sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )+2 \left (\sin ^{3}\left (2 b x +2 a \right )\right )-2 \sin \left (2 b x +2 a \right )\right )}{24 \cos \left (2 b x +2 a \right ) \sqrt {\sin \left (2 b x +2 a \right )}}\right )}{b} \]
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 \[ \int \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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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