Integrand size = 14, antiderivative size = 66 \[ \int (a+a \sin (c+d x))^{2/3} \, dx=-\frac {2 \sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{d (1+\sin (c+d x))^{7/6}} \] Output:
-2*2^(1/6)*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)/d/(1+sin(d*x+c))^(7/6)
Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.88 \[ \int (a+a \sin (c+d x))^{2/3} \, dx=-\frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {2-2 \sin (c+d x)}\right ) (a (1+\sin (c+d x)))^{2/3}}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {2-2 \sin (c+d x)}} \] Input:
Integrate[(a + a*Sin[c + d*x])^(2/3),x]
Output:
(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[2 - 2*Sin[c + d*x]])*(a*(1 + Sin[ c + d*x]))^(2/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[2 - 2*Si n[c + d*x]])
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {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.
\(\displaystyle \int (a \sin (c+d x)+a)^{2/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (c+d x)+a)^{2/3}dx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {(a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{(\sin (c+d x)+1)^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{(\sin (c+d x)+1)^{2/3}}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle -\frac {2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}}\) |
Input:
Int[(a + a*Sin[c + d*x])^(2/3),x]
Output:
(-2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d* x])/2]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))
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]
\[\int \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
Input:
int((a+a*sin(d*x+c))^(2/3),x)
Output:
int((a+a*sin(d*x+c))^(2/3),x)
\[ \int (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^(2/3),x, algorithm="fricas")
Output:
integral((a*sin(d*x + c) + a)^(2/3), x)
\[ \int (a+a \sin (c+d x))^{2/3} \, dx=\int \left (a \sin {\left (c + d x \right )} + a\right )^{\frac {2}{3}}\, dx \] Input:
integrate((a+a*sin(d*x+c))**(2/3),x)
Output:
Integral((a*sin(c + d*x) + a)**(2/3), x)
\[ \int (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^(2/3),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) + a)^(2/3), x)
\[ \int (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*sin(d*x+c))^(2/3),x, algorithm="giac")
Output:
integrate((a*sin(d*x + c) + a)^(2/3), x)
Timed out. \[ \int (a+a \sin (c+d x))^{2/3} \, dx=\int {\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3} \,d x \] Input:
int((a + a*sin(c + d*x))^(2/3),x)
Output:
int((a + a*sin(c + d*x))^(2/3), x)
\[ \int (a+a \sin (c+d x))^{2/3} \, dx=a^{\frac {2}{3}} \left (\int \left (\sin \left (d x +c \right )+1\right )^{\frac {2}{3}}d x \right ) \] Input:
int((a+a*sin(d*x+c))^(2/3),x)
Output:
a**(2/3)*int((sin(c + d*x) + 1)**(2/3),x)