Optimal. Leaf size=126 \[ -\frac{19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d} \]
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Rubi [A] time = 0.145496, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2759, 2751, 2652, 2651} \[ -\frac{19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d} \]
Antiderivative was successfully verified.
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Rule 2759
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=-\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac{3 \int \left (\frac{5 a}{3}-a \sin (c+d x)\right ) (a+a \sin (c+d x))^{2/3} \, dx}{8 a}\\ &=\frac{9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac{19}{40} \int (a+a \sin (c+d x))^{2/3} \, dx\\ &=\frac{9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac{\left (19 (a+a \sin (c+d x))^{2/3}\right ) \int (1+\sin (c+d x))^{2/3} \, dx}{40 (1+\sin (c+d x))^{2/3}}\\ &=\frac{9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac{19 \cos (c+d x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{10\ 2^{5/6} d (1+\sin (c+d x))^{7/6}}-\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}\\ \end{align*}
Mathematica [A] time = 0.438797, size = 151, normalized size = 1.2 \[ \frac{3 (a (\sin (c+d x)+1))^{2/3} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (19 \sqrt{2} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )+\sqrt{1-\sin (c+d x)} (5 \cos (2 (c+d x))-14 (\sin (c+d x)+2))\right )}{80 d \sqrt{1-\sin (c+d x)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.197, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}\, dx \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{2}{3}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}}, 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 \left (a \left (\sin{\left (c + d x \right )} + 1\right )\right )^{\frac{2}{3}} \sin ^{2}{\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{2}{3}} \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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