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

Optimal. Leaf size=65 \[ \frac{\sin (c+d x) \sqrt [6]{\cos (c+d x)+1} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{\sqrt [6]{2} d (a \cos (c+d x)+a)^{2/3}} \]

[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))

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Rubi [A]  time = 0.0312535, 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} \[ \frac{\sin (c+d x) \sqrt [6]{\cos (c+d x)+1} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{\sqrt [6]{2} d (a \cos (c+d x)+a)^{2/3}} \]

Antiderivative was successfully verified.

[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 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}{(a+a \cos (c+d x))^{2/3}} \, dx &=\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)} \, _2F_1\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]  time = 0.0561379, size = 67, normalized size = 1.03 \[ \frac{6 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{d (a (\cos (c+d x)+1))^{2/3}} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( a+\cos \left ( dx+c \right ) a \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cos{\left (c + d x \right )} + a\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)