Integrand size = 21, antiderivative size = 96 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{5 d}-\frac {4 \sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{5 d (1+\sin (c+d x))^{7/6}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 96, 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} \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {4 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d (\sin (c+d x)+1)^{7/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d} \]
[In]
[Out]
Rule 2730
Rule 2731
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{5 d}+\frac {2}{5} \int (a+a \sin (c+d x))^{2/3} \, dx \\ & = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{5 d}+\frac {\left (2 (a+a \sin (c+d x))^{2/3}\right ) \int (1+\sin (c+d x))^{2/3} \, dx}{5 (1+\sin (c+d x))^{2/3}} \\ & = -\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{5 d}-\frac {4 \sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{5 d (1+\sin (c+d x))^{7/6}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sin (c+d x)))^{2/3} \left (-\sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {1-\sin (c+d x)} (2+\sin (c+d x))\right )}{5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {1-\sin (c+d x)}} \]
[In]
[Out]
\[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right ) \,d x } \]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \sin {\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right ) \,d x } \]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3} \,d x \]
[In]
[Out]