Integrand size = 23, antiderivative size = 127 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {37\ 2^{5/6} a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{35 d (1+\sin (c+d x))^{5/6}}+\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{70 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{10 a d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 127, 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 = {2838, 2830, 2731, 2730} \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=-\frac {37\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{35 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{10 a d}+\frac {9 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{70 d} \]
[In]
[Out]
Rule 2730
Rule 2731
Rule 2830
Rule 2838
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{10 a d}+\frac {3 \int \left (\frac {7 a}{3}-a \sin (c+d x)\right ) (a+a \sin (c+d x))^{4/3} \, dx}{10 a} \\ & = \frac {9 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{70 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{10 a d}+\frac {37}{70} \int (a+a \sin (c+d x))^{4/3} \, dx \\ & = \frac {9 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{70 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{10 a d}+\frac {\left (37 a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{70 \sqrt [3]{1+\sin (c+d x)}} \\ & = -\frac {37\ 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)}}{35 d (1+\sin (c+d x))^{5/6}}+\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{70 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{10 a d} \\ \end{align*}
Time = 3.64 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.97 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\frac {(a (1+\sin (c+d x)))^{4/3} \left (-\frac {3 (-185+60 \cos (c+d x)-7 \cos (3 (c+d x))+22 \sin (2 (c+d x)))}{20 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {37 \left (-2 \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{6};\frac {5}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \left (2 \cos \left (\frac {1}{4} (2 c+\pi +2 d x)\right )+3 \sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )\right )}{2\ 2^{2/3} \sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{8/3} \sqrt [3]{\sin \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}}\right )}{14 d} \]
[In]
[Out]
\[\int \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}} \sin ^{2}{\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx=\int {\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \]
[In]
[Out]