Optimal. Leaf size=124 \[ -\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d} \]
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Rubi [A] time = 0.255076, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2856, 2674, 2673} \[ -\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2856
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac{2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac{1}{3} \int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac{2 a \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{21 d}-\frac{2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac{1}{21} (8 a) \int \cos ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{21 d}-\frac{2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac{1}{105} \left (32 a^2\right ) \int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a+a \sin (c+d x))^{3/2}}-\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{21 d}-\frac{2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}\\ \end{align*}
Mathematica [A] time = 1.45006, size = 100, normalized size = 0.81 \[ \frac{a \sqrt{a (\sin (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (-741 \sin (c+d x)+35 \sin (3 (c+d x))+240 \cos (2 (c+d x))-664)}{630 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.75, size = 77, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 35\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+120\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+159\,\sin \left ( dx+c \right ) +106 \right ) }{315\,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}} \cos \left (d x + c\right )^{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 = 1.5655, size = 396, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (35 \, a \cos \left (d x + c\right )^{5} - 50 \, a \cos \left (d x + c\right )^{4} - 109 \, a \cos \left (d x + c\right )^{3} + 8 \, a \cos \left (d x + c\right )^{2} - 32 \, a \cos \left (d x + c\right ) -{\left (35 \, a \cos \left (d x + c\right )^{4} + 85 \, a \cos \left (d x + c\right )^{3} - 24 \, a \cos \left (d x + c\right )^{2} - 32 \, a \cos \left (d x + c\right ) - 64 \, a\right )} \sin \left (d x + c\right ) - 64 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{315 \,{\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(-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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{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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