Integrand size = 21, antiderivative size = 97 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {8\ 2^{5/6} a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{7 d (1+\sin (c+d x))^{5/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d} \]
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Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2731, 2730} \[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {8\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{7 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{7 d} \]
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Rule 2730
Rule 2731
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d}+\frac {4}{7} \int (a+a \sin (c+d x))^{4/3} \, dx \\ & = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d}+\frac {\left (4 a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{7 \sqrt [3]{1+\sin (c+d x)}} \\ & = -\frac {8\ 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)}}{7 d (1+\sin (c+d x))^{5/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(97)=194\).
Time = 2.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.41 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\frac {(a (1+\sin (c+d x)))^{4/3} \left (40 \sqrt [3]{2} \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{6};\frac {5}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )-\sqrt {2-2 \sin (c+d x)} \left (20 \sqrt [3]{2} \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right )+3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{2/3} (4 \cos (c+d x)+\sin (2 (c+d x))) \sqrt [3]{\sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}\right )\right )}{28 d \sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{8/3} \sqrt [3]{\sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}} \]
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\[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right ) \,d x } \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}} \sin {\left (c + d x \right )}\, dx \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right ) \,d x } \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \]
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