3.11 \(\int \frac{1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx\)

Optimal. Leaf size=65 \[ \frac{\sqrt [6]{2} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}} \]

[Out]

(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))

________________________________________________________________________________________

Rubi [A]  time = 0.0316252, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ \frac{\sqrt [6]{2} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(-1/3),x]

[Out]

(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))

Rule 2652

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart
[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2651

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(2^(n + 1/2)*a^(n - 1/2)*b*Cos[c + d*x]*Hy
pergeometric2F1[1/2, 1/2 - n, 3/2, (1*(1 - (b*Sin[c + d*x])/a))/2])/(d*Sqrt[a + b*Sin[c + d*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*} \int \frac{1}{\sqrt [3]{a+a \cos (c+d x)}} \, dx &=\frac{\sqrt [3]{1+\cos (c+d x)} \int \frac{1}{\sqrt [3]{1+\cos (c+d x)}} \, dx}{\sqrt [3]{a+a \cos (c+d x)}}\\ &=\frac{\sqrt [6]{2} \, _2F_1\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)}}\\ \end{align*}

Mathematica [A]  time = 0.0544177, size = 67, normalized size = 1.03 \[ -\frac{6 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{d \sqrt [3]{a (\cos (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^(-1/3),x]

[Out]

(-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))

________________________________________________________________________________________

Maple [F]  time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{a+\cos \left ( dx+c \right ) a}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*cos(d*x + c) + a)^(-1/3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{a \cos{\left (c + d x \right )} + a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*cos(c + d*x) + a)**(-1/3), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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