Optimal. Leaf size=128 \[ -\frac{2 a^2 c \cos ^3(c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{8 a^3 c \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 c \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}+\frac{4 a c \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d} \]
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Rubi [A] time = 0.347159, antiderivative size = 165, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {2976, 2981, 2759, 2751, 2646} \[ \frac{2 a^2 c \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 c \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}+\frac{2 a c \sin ^3(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{9 d}-\frac{2 c \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{21 d}+\frac{4 a c \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{63 d} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2981
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} (c-c \sin (c+d x)) \, dx &=\frac{2 a c \cos (c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{9 d}+\frac{2}{9} \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{3 a c}{2}-\frac{1}{2} a c \sin (c+d x)\right ) \, dx\\ &=\frac{2 a^2 c \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}+\frac{2 a c \cos (c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{9 d}+\frac{1}{21} (5 a c) \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{2 a^2 c \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}+\frac{2 a c \cos (c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{9 d}-\frac{2 c \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{21 d}+\frac{1}{21} (2 c) \int \left (\frac{3 a}{2}-a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{2 a^2 c \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}+\frac{4 a c \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{63 d}+\frac{2 a c \cos (c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{9 d}-\frac{2 c \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{21 d}+\frac{1}{9} (a c) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 a^2 c \cos (c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{2 a^2 c \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}+\frac{4 a c \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{63 d}+\frac{2 a c \cos (c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{9 d}-\frac{2 c \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{21 d}\\ \end{align*}
Mathematica [A] time = 0.845021, size = 101, normalized size = 0.79 \[ \frac{a c \sqrt{a (\sin (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (-69 \sin (c+d x)+7 \sin (3 (c+d x))+30 \cos (2 (c+d x))-62)}{126 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.753, size = 78, 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}c \left ( 7\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+15\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+12\,\sin \left ( dx+c \right ) +8 \right ) }{63\,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}}{\left (c \sin \left (d x + c\right ) - c\right )} \sin \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98642, size = 405, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (7 \, a c \cos \left (d x + c\right )^{5} - a c \cos \left (d x + c\right )^{4} - 11 \, a c \cos \left (d x + c\right )^{3} + a c \cos \left (d x + c\right )^{2} - 4 \, a c \cos \left (d x + c\right ) - 8 \, a c -{\left (7 \, a c \cos \left (d x + c\right )^{4} + 8 \, a c \cos \left (d x + c\right )^{3} - 3 \, a c \cos \left (d x + c\right )^{2} - 4 \, a c \cos \left (d x + c\right ) - 8 \, a c\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{63 \,{\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}}{\left (c \sin \left (d x + c\right ) - c\right )} \sin \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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