Optimal. Leaf size=53 \[ \frac{\sin (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.0395403, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4304, 4291} \[ \frac{\sin (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 4304
Rule 4291
Rubi steps
\begin{align*} \int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=\frac{\sin (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{2}{3} \int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=\frac{\sin (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.102788, size = 43, normalized size = 0.81 \[ \frac{\sqrt{\sin (2 (a+b x))} \left (\frac{1}{12} \tan (a+b x) \sec (a+b x)-\frac{1}{4} \csc (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 18.474, size = 597, normalized size = 11.3 \begin{align*} \text{result too large to display} \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 )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.509805, size = 184, normalized size = 3.47 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) + \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{12 \, b \cos \left (b x + a\right )^{2} \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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