Integrand size = 14, antiderivative size = 65 \[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\frac {\sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d \sqrt [6]{1+\cos (c+d x)} \sqrt [3]{a+a \cos (c+d x)}} \] Output:
2^(1/6)*hypergeom([1/2, 5/6],[3/2],1/2-1/2*cos(d*x+c))*sin(d*x+c)/d/(1+cos (d*x+c))^(1/6)/(a+a*cos(d*x+c))^(1/3)
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=-\frac {6 \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt [3]{a (1+\cos (c+d x))}} \] Input:
Integrate[(a + a*Cos[c + d*x])^(-1/3),x]
Output:
(-6*Cot[(c + d*x)/2]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[(c + d*x)/2]^2]* Sqrt[Sin[(c + d*x)/2]^2])/(d*(a*(1 + Cos[c + d*x]))^(1/3))
Time = 0.28 (sec) , antiderivative size = 65, 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.
\(\displaystyle \int \frac {1}{\sqrt [3]{a \cos (c+d x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt [3]{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle \frac {\sqrt [3]{\cos (c+d x)+1} \int \frac {1}{\sqrt [3]{\cos (c+d x)+1}}dx}{\sqrt [3]{a \cos (c+d x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt [3]{\cos (c+d x)+1} \int \frac {1}{\sqrt [3]{\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sqrt [3]{a \cos (c+d x)+a}}\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle \frac {\sqrt [6]{2} \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}}\) |
Input:
Int[(a + a*Cos[c + d*x])^(-1/3),x]
Output:
(2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d* x])/(d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))
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]
\[\int \frac {1}{\left (a +a \cos \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
Input:
int(1/(a+a*cos(d*x+c))^(1/3),x)
Output:
int(1/(a+a*cos(d*x+c))^(1/3),x)
\[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*cos(d*x+c))^(1/3),x, algorithm="fricas")
Output:
integral((a*cos(d*x + c) + a)^(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{a \cos {\left (c + d x \right )} + a}}\, dx \] Input:
integrate(1/(a+a*cos(d*x+c))**(1/3),x)
Output:
Integral((a*cos(c + d*x) + a)**(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*cos(d*x+c))^(1/3),x, algorithm="maxima")
Output:
integrate((a*cos(d*x + c) + a)^(-1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(a+a*cos(d*x+c))^(1/3),x, algorithm="giac")
Output:
integrate((a*cos(d*x + c) + a)^(-1/3), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\int \frac {1}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \] Input:
int(1/(a + a*cos(c + d*x))^(1/3),x)
Output:
int(1/(a + a*cos(c + d*x))^(1/3), x)
\[ \int \frac {1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=\frac {\int \frac {1}{\left (\cos \left (d x +c \right )+1\right )^{\frac {1}{3}}}d x}{a^{\frac {1}{3}}} \] Input:
int(1/(a+a*cos(d*x+c))^(1/3),x)
Output:
int(1/(cos(c + d*x) + 1)**(1/3),x)/a**(1/3)