Integrand size = 23, antiderivative size = 129 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}+\frac {13 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2837, 2830, 2731, 2730} \[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\frac {13 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a \sin (c+d x)+a}}-\frac {3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}} \]
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Rule 2730
Rule 2731
Rule 2830
Rule 2837
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}+\frac {3 \int \frac {-\frac {4 a}{3}+\frac {5}{3} a \sin (c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{5 a^2} \\ & = -\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}-\frac {13 \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{10 a} \\ & = -\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}-\frac {\left (13 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac {1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{10 a \sqrt [3]{a+a \sin (c+d x)}} \\ & = -\frac {3 \cos (c+d x)}{5 d (a+a \sin (c+d x))^{4/3}}-\frac {3 \cos (c+d x)}{2 a d \sqrt [3]{a+a \sin (c+d x)}}+\frac {13 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {3 \cos (c+d x) \left (13 \sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right ) (1+\sin (c+d x))+\sqrt {1-\sin (c+d x)} (7+5 \sin (c+d x))\right )}{10 d \sqrt {1-\sin (c+d x)} (a (1+\sin (c+d x)))^{4/3}} \]
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\[\int \frac {\sin ^{2}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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