Optimal. Leaf size=86 \[ -\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.062433, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2751, 2647, 2646} \[ -\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{3}{5} \int (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{5} (4 a) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.15917, size = 115, normalized size = 1.34 \[ -\frac{(a (\sin (c+d x)+1))^{3/2} \left (-20 \sin \left (\frac{1}{2} (c+d x)\right )+5 \sin \left (\frac{3}{2} (c+d x)\right )+\sin \left (\frac{5}{2} (c+d x)\right )+20 \cos \left (\frac{1}{2} (c+d x)\right )+5 \cos \left (\frac{3}{2} (c+d x)\right )-\cos \left (\frac{5}{2} (c+d x)\right )\right )}{10 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.437, size = 63, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{2}+3\,\sin \left ( dx+c \right ) +6 \right ) }{5\,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}} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52276, size = 259, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right )^{2} + 3 \, 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}}{5 \,{\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 \left (\sin{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}} \sin{\left (c + d 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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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