Integrand size = 23, antiderivative size = 126 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac {19 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{10\ 2^{5/6} d (1+\sin (c+d x))^{7/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d} \]
9/40*cos(d*x+c)*(a+a*sin(d*x+c))^(2/3)/d-19/20*cos(d*x+c)*hypergeom([-1/6, 1/2],[3/2],1/2-1/2*sin(d*x+c))*(a+a*sin(d*x+c))^(2/3)*2^(1/6)/d/(1+sin(d* x+c))^(7/6)-3/8*cos(d*x+c)*(a+a*sin(d*x+c))^(5/3)/a/d
Time = 0.86 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.20 \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sin (c+d x)))^{2/3} \left (19 \sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {1-\sin (c+d x)} (5 \cos (2 (c+d x))-14 (2+\sin (c+d x)))\right )}{80 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {1-\sin (c+d x)}} \]
(3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(2/3)*(19* Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sq rt[1 - Sin[c + d*x]]*(5*Cos[2*(c + d*x)] - 14*(2 + Sin[c + d*x]))))/(80*d* (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[1 - Sin[c + d*x]])
Time = 0.56 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 3238, 27, 3042, 3230, 3042, 3131, 3042, 3130}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^2(c+d x) (a \sin (c+d x)+a)^{2/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^2 (a \sin (c+d x)+a)^{2/3}dx\) |
\(\Big \downarrow \) 3238 |
\(\displaystyle \frac {3 \int \frac {1}{3} (5 a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{2/3}dx}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int (5 a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{2/3}dx}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (5 a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{2/3}dx}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle \frac {\frac {19}{5} a \int (\sin (c+d x) a+a)^{2/3}dx+\frac {9 a \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {19}{5} a \int (\sin (c+d x) a+a)^{2/3}dx+\frac {9 a \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {\frac {19 a (a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{5 (\sin (c+d x)+1)^{2/3}}+\frac {9 a \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {19 a (a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{5 (\sin (c+d x)+1)^{2/3}}+\frac {9 a \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle \frac {\frac {9 a \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}-\frac {38 \sqrt [6]{2} a \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d (\sin (c+d x)+1)^{7/6}}}{8 a}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}\) |
(-3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/3))/(8*a*d) + ((9*a*Cos[c + d*x]* (a + a*Sin[c + d*x])^(2/3))/(5*d) - (38*2^(1/6)*a*Cos[c + d*x]*Hypergeomet ric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(2/3))/( 5*d*(1 + Sin[c + d*x])^(7/6)))/(8*a)
3.1.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] ) Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2 ))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*Si n[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && ! LtQ[m, -2^(-1)]
\[\int \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \sin ^{2}{\left (c + d x \right )}\, dx \]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
\[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int {\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3} \,d x \]