Optimal. Leaf size=47 \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b}-\frac{2 \sqrt{\sin (a+b x)} \cos (a+b x)}{3 b} \]
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Rubi [A] time = 0.0164747, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b}-\frac{2 \sqrt{\sin (a+b x)} \cos (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sin ^{\frac{3}{2}}(a+b x) \, dx &=-\frac{2 \cos (a+b x) \sqrt{\sin (a+b x)}}{3 b}+\frac{1}{3} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac{2 \cos (a+b x) \sqrt{\sin (a+b x)}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0378158, size = 40, normalized size = 0.85 \[ -\frac{2 \left (F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )+\sqrt{\sin (a+b x)} \cos (a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 88, normalized size = 1.9 \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) } \left ({\frac{1}{3}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) }{3}} \right ){\frac{1}{\sqrt{\sin \left ( bx+a \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 \sin \left (b x + a\right )^{\frac{3}{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 (\sin \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{\frac{3}{2}}{\left (a + b x \right )}\, dx \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 \sin \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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