Integrand size = 23, antiderivative size = 162 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {388\ 2^{5/6} a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{455 d (1+\sin (c+d x))^{5/6}}-\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d} \]
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Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2862, 3047, 3102, 2830, 2731, 2730} \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {388\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{455 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac {72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d} \]
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Rule 2730
Rule 2731
Rule 2830
Rule 2862
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}+\frac {3 \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \left (2 a+\frac {4}{3} a \sin (c+d x)\right ) \, dx}{13 a} \\ & = -\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}+\frac {3 \int (a+a \sin (c+d x))^{4/3} \left (2 a \sin (c+d x)+\frac {4}{3} a \sin ^2(c+d x)\right ) \, dx}{13 a} \\ & = -\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {9 \int (a+a \sin (c+d x))^{4/3} \left (\frac {28 a^2}{9}+\frac {16}{3} a^2 \sin (c+d x)\right ) \, dx}{130 a^2} \\ & = -\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {194}{455} \int (a+a \sin (c+d x))^{4/3} \, dx \\ & = -\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {\left (194 a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{455 \sqrt [3]{1+\sin (c+d x)}} \\ & = -\frac {388\ 2^{5/6} a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{455 d (1+\sin (c+d x))^{5/6}}-\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.59 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\frac {(a (1+\sin (c+d x)))^{4/3} \left (\frac {3 (1940-790 \cos (c+d x)+98 \cos (3 (c+d x))-278 \sin (2 (c+d x))+35 \sin (4 (c+d x)))}{40 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {97 \left (-2 \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{6};\frac {5}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \left (2 \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right )+3 \sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )\right )}{2^{2/3} \sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{8/3} \sqrt [3]{\sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}}\right )}{91 d} \]
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\[\int \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
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\[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\text {Timed out} \]
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\[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{3} \,d x } \]
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\[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int {\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \]
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