\(\int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx\) [12]

Optimal result

Integrand size = 14, antiderivative size = 65 \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {\sqrt [6]{1+\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}} \]

[Out]

1/2*(1+cos(d*x+c))^(1/6)*hypergeom([1/2, 7/6],[3/2],1/2-1/2*cos(d*x+c))*sin(d*x+c)*2^(5/6)/d/(a+a*cos(d*x+c))^
(2/3)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2731, 2730} \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {\sin (c+d x) \sqrt [6]{\cos (c+d x)+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{\sqrt [6]{2} d (a \cos (c+d x)+a)^{2/3}} \]

[In]

Int[(a + a*Cos[c + d*x])^(-2/3),x]

[Out]

((1 + Cos[c + d*x])^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(2^(1/6)*d*(a +
a*Cos[c + d*x])^(2/3))

Rule 2730

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]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(1+\cos (c+d x))^{2/3} \int \frac {1}{(1+\cos (c+d x))^{2/3}} \, dx}{(a+a \cos (c+d x))^{2/3}} \\ & = \frac {\sqrt [6]{1+\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {6 \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d (a (1+\cos (c+d x)))^{2/3}} \]

[In]

Integrate[(a + a*Cos[c + d*x])^(-2/3),x]

[Out]

(6*Cot[(c + d*x)/2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d*(a*(1 +
Cos[c + d*x]))^(2/3))

Maple [F]

\[\int \frac {1}{\left (a +\cos \left (d x +c \right ) a \right )^{\frac {2}{3}}}d x\]

[In]

int(1/(a+cos(d*x+c)*a)^(2/3),x)

[Out]

int(1/(a+cos(d*x+c)*a)^(2/3),x)

Fricas [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(a+a*cos(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

integral((a*cos(d*x + c) + a)^(-2/3), x)

Sympy [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (a \cos {\left (c + d x \right )} + a\right )^{\frac {2}{3}}}\, dx \]

[In]

integrate(1/(a+a*cos(d*x+c))**(2/3),x)

[Out]

Integral((a*cos(c + d*x) + a)**(-2/3), x)

Maxima [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(a+a*cos(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + a)^(-2/3), x)

Giac [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(a+a*cos(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + a)^(-2/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]

[In]

int(1/(a + a*cos(c + d*x))^(2/3),x)

[Out]

int(1/(a + a*cos(c + d*x))^(2/3), x)