Optimal. Leaf size=106 \[ \frac{30 \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{77 b}-\frac{30 \cos (2 a+2 b x)}{77 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{18 \cos (2 a+2 b x)}{77 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{11 b \sin ^{\frac{7}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.059475, antiderivative size = 106, normalized size of antiderivative = 1., 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 4300
Rule 2636
Rule 2641
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*}
Mathematica [A] time = 0.343713, size = 86, normalized size = 0.81 \[ \frac{480 \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )+\sqrt{\sin (2 (a+b x))} \left (-7 \csc ^6(a+b x)-32 \csc ^4(a+b x)-141 \csc ^2(a+b x)+11 \sec ^2(a+b x) \left (\sec ^2(a+b x)+9\right )\right )}{1232 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 47.27, size = 167, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2}}{64\,b} \left ( -{\frac{64\,\sqrt{2}}{11} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{11}{2}}}}+{\frac{32\,\sqrt{2}}{77\,\cos \left ( 2\,bx+2\,a \right ) } \left ( 15\,\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{5}+30\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{6}-12\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}-4\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{2}-14 \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{11}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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