Optimal. Leaf size=162 \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt [3]{a \sin (c+d x)+a}}+\frac{6 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}} \]
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Rubi [A] time = 0.26968, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2783, 2968, 3019, 2751, 2652, 2651} \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt [3]{a \sin (c+d x)+a}}+\frac{6 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2783
Rule 2968
Rule 3019
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx &=-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac{3 \int \frac{\sin (c+d x) \left (2 a-\frac{4}{3} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{4/3}} \, dx}{5 a}\\ &=-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac{3 \int \frac{2 a \sin (c+d x)-\frac{4}{3} a \sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx}{5 a}\\ &=\frac{6 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac{9 \int \frac{-\frac{40 a^2}{9}+\frac{20}{9} a^2 \sin (c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{25 a^3}\\ &=\frac{6 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt [3]{a+a \sin (c+d x)}}+\frac{2 \int \frac{1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{a}\\ &=\frac{6 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt [3]{a+a \sin (c+d x)}}+\frac{\left (2 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac{1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=\frac{6 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt [3]{a+a \sin (c+d x)}}-\frac{2 \sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.483215, size = 116, normalized size = 0.72 \[ \frac{3 \cos (c+d x) \left (20 \sqrt{2} (\sin (c+d x)+1) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )+\sqrt{1-\sin (c+d x)} (4 \sin (c+d x)+\cos (2 (c+d x))+7)\right )}{10 d \sqrt{1-\sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.284, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{-{\frac{4}{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 \frac{\sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{3}}}\,{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 (\frac{{\left (\cos \left (d x + c\right )^{2} - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \sin \left (d x + c\right )}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\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 \frac{\sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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