Optimal. Leaf size=106 \[ -\frac{14 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{15 b}-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{14 \cos (2 a+2 b x)}{15 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.0580496, 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, 2639} \[ -\frac{14 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{15 b}-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{14 \cos (2 a+2 b x)}{15 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx &=-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{14}{9} \int \frac{1}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{14}{15} \int \frac{1}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{14 \cos (2 a+2 b x)}{15 b \sqrt{\sin (2 a+2 b x)}}-\frac{14}{15} \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=-\frac{14 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{15 b}-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{14 \cos (2 a+2 b x)}{15 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.814174, size = 85, normalized size = 0.8 \[ -\frac{336 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )+\frac{(98 \cos (2 (a+b x))-28 \cos (4 (a+b x))-42 \cos (6 (a+b x))+21 \cos (8 (a+b x))-9) \csc ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 (a+b x))}}{360 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 30.493, size = 240, normalized size = 2.3 \begin{align*}{\frac{\sqrt{2}}{32\,b} \left ( -{\frac{32\,\sqrt{2}}{9} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{9}{2}}}}+{\frac{16\,\sqrt{2}}{45\,\cos \left ( 2\,bx+2\,a \right ) } \left ( 42\,\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 ) } \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}{\it EllipticE} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) -21\,\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 ) } \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) +42\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{6}-28\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}-4\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{2}-10 \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{9}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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{7}{2}}}\,{d x} \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} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}}{\cos \left (2 \, b x + 2 \, a\right )^{4} - 2 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 1}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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