Optimal. Leaf size=59 \[ -\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.0292287, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^{3/2} \, dx &=-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.131385, size = 89, normalized size = 1.51 \[ -\frac{(a (\sin (c+d x)+1))^{3/2} \left (-9 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )+9 \cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )}{3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 53, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( \sin \left ( dx+c \right ) +5 \right ) }{3\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64163, size = 204, normalized size = 3.46 \begin{align*} -\frac{2 \,{\left (a \cos \left (d x + c\right )^{2} + 5 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right ) - 4 \, a\right )} \sin \left (d x + c\right ) + 4 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\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 \left (a \sin{\left (c + d x \right )} + a\right )^{\frac{3}{2}}\, 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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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