Optimal. Leaf size=81 \[ \frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2 \sin (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{4 \cos (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.0691508, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4296, 4304, 4291} \[ \frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2 \sin (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{4 \cos (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 4296
Rule 4304
Rule 4291
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx &=\frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2}{7} \int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=\frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2 \sin (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4}{21} \int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=\frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}+\frac{2 \sin (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{4 \cos (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.106691, size = 55, normalized size = 0.68 \[ -\frac{\sqrt{\sin (2 (a+b x))} (12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc (a+b x) \sec ^4(a+b x)}{336 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{9}{2}}}}\, dx \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{\sin \left (b x + a\right )^{3}}{\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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Fricas [A] time = 0.517084, size = 213, normalized size = 2.63 \begin{align*} -\frac{32 \, \cos \left (b x + a\right )^{4} \sin \left (b x + a\right ) + \sqrt{2}{\left (32 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{336 \, b \cos \left (b x + a\right )^{4} \sin \left (b x + a\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(-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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