Integrand size = 14, antiderivative size = 66 \[ \int (a+a \cos (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} (a+a \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \] Output:
2*2^(1/6)*(a+a*cos(d*x+c))^(2/3)*hypergeom([-1/6, 1/2],[3/2],1/2-1/2*cos(d *x+c))*sin(d*x+c)/d/(1+cos(d*x+c))^(7/6)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int (a+a \cos (c+d x))^{2/3} \, dx=-\frac {6 (a (1+\cos (c+d x)))^{2/3} \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{7 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^(2/3),x]
Output:
(-6*(a*(1 + Cos[c + d*x]))^(2/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 7 /6, 13/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(7*d)
Time = 0.28 (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.
\(\displaystyle \int (a \cos (c+d x)+a)^{2/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{2/3}dx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {(a \cos (c+d x)+a)^{2/3} \int (\cos (c+d x)+1)^{2/3}dx}{(\cos (c+d x)+1)^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(a \cos (c+d x)+a)^{2/3} \int \left (\sin \left (c+d x+\frac {\pi }{2}\right )+1\right )^{2/3}dx}{(\cos (c+d x)+1)^{2/3}}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle \frac {2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}}\) |
Input:
Int[(a + a*Cos[c + d*x])^(2/3),x]
Output:
(2*2^(1/6)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(d*(1 + Cos[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]
\[\int \left (a +a \cos \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
Input:
int((a+a*cos(d*x+c))^(2/3),x)
Output:
int((a+a*cos(d*x+c))^(2/3),x)
\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(2/3),x, algorithm="fricas")
Output:
integral((a*cos(d*x + c) + a)^(2/3), x)
\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {2}{3}}\, dx \] Input:
integrate((a+a*cos(d*x+c))**(2/3),x)
Output:
Integral((a*cos(c + d*x) + a)**(2/3), x)
\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(2/3),x, algorithm="maxima")
Output:
integrate((a*cos(d*x + c) + a)^(2/3), x)
\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^(2/3),x, algorithm="giac")
Output:
integrate((a*cos(d*x + c) + a)^(2/3), x)
Timed out. \[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \] Input:
int((a + a*cos(c + d*x))^(2/3),x)
Output:
int((a + a*cos(c + d*x))^(2/3), x)
\[ \int (a+a \cos (c+d x))^{2/3} \, dx=a^{\frac {2}{3}} \left (\int \left (\cos \left (d x +c \right )+1\right )^{\frac {2}{3}}d x \right ) \] Input:
int((a+a*cos(d*x+c))^(2/3),x)
Output:
a**(2/3)*int((cos(c + d*x) + 1)**(2/3),x)