3.84 \(\int \frac{1}{1-\cos ^6(x)} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\cot (x)}{3}-\frac{\tan ^{-1}\left (\sqrt{1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1+\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1-(-1)^{2/3}}} \]

[Out]

-ArcTan[Sqrt[1 + (-1)^(1/3)]*Cot[x]]/(3*Sqrt[1 + (-1)^(1/3)]) - ArcTan[Sqrt[1 - (-1)^(2/3)]*Cot[x]]/(3*Sqrt[1
- (-1)^(2/3)]) - Cot[x]/3

________________________________________________________________________________________

Rubi [A]  time = 0.118622, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3211, 3181, 203, 3175, 3767, 8} \[ -\frac{\cot (x)}{3}-\frac{\tan ^{-1}\left (\sqrt{1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1+\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1-(-1)^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x]^6)^(-1),x]

[Out]

-ArcTan[Sqrt[1 + (-1)^(1/3)]*Cot[x]]/(3*Sqrt[1 + (-1)^(1/3)]) - ArcTan[Sqrt[1 - (-1)^(2/3)]*Cot[x]]/(3*Sqrt[1
- (-1)^(2/3)]) - Cot[x]/3

Rule 3211

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{1-\cos ^6(x)} \, dx &=\frac{1}{3} \int \frac{1}{1-\cos ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1+\sqrt [3]{-1} \cos ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1-(-1)^{2/3} \cos ^2(x)} \, dx\\ &=\frac{1}{3} \int \csc ^2(x) \, dx-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1+\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1-(-1)^{2/3}}}-\frac{1}{3} \operatorname{Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\frac{\tan ^{-1}\left (\sqrt{1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1+\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1-(-1)^{2/3}}}-\frac{\cot (x)}{3}\\ \end{align*}

Mathematica [C]  time = 0.264715, size = 117, normalized size = 1.65 \[ \frac{\sin (x) (8 \cos (2 x)+\cos (4 x)+15) \left (6 \cos (x)+i \sqrt [4]{-3} \left (\sqrt{3}+3 i\right ) \sin (x) \tan ^{-1}\left (\frac{1}{2} \sqrt [4]{-\frac{1}{3}} \left (\sqrt{3}-i\right ) \tan (x)\right )+\sqrt [4]{-3} \left (\sqrt{3}-3 i\right ) \sin (x) \tan ^{-1}\left (\frac{(-1)^{3/4} \left (\sqrt{3}+i\right ) \tan (x)}{2 \sqrt [4]{3}}\right )\right )}{144 \left (\cos ^6(x)-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cos[x]^6)^(-1),x]

[Out]

((15 + 8*Cos[2*x] + Cos[4*x])*Sin[x]*(6*Cos[x] + I*(-3)^(1/4)*(3*I + Sqrt[3])*ArcTan[((-1/3)^(1/4)*(-I + Sqrt[
3])*Tan[x])/2]*Sin[x] + (-3)^(1/4)*(-3*I + Sqrt[3])*ArcTan[((-1)^(3/4)*(I + Sqrt[3])*Tan[x])/(2*3^(1/4))]*Sin[
x]))/(144*(-1 + Cos[x]^6))

________________________________________________________________________________________

Maple [B]  time = 0.092, size = 233, normalized size = 3.3 \begin{align*} -{\frac{1}{3\,\tan \left ( x \right ) }}-{\frac{\sqrt{3}\sqrt{2\,\sqrt{3}-3}\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+\tan \left ( x \right ) \sqrt{2\,\sqrt{3}-3}+\sqrt{3} \right ) }{36}}+{\frac{1}{3\,\sqrt{2\,\sqrt{3}+3}}\arctan \left ({\frac{2\,\tan \left ( x \right ) +\sqrt{2\,\sqrt{3}-3}}{\sqrt{2\,\sqrt{3}+3}}} \right ) }+{\frac{\sqrt{3}}{6\,\sqrt{2\,\sqrt{3}+3}}\arctan \left ({\frac{2\,\tan \left ( x \right ) +\sqrt{2\,\sqrt{3}-3}}{\sqrt{2\,\sqrt{3}+3}}} \right ) }+{\frac{\sqrt{3}\sqrt{2\,\sqrt{3}-3}\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}-\tan \left ( x \right ) \sqrt{2\,\sqrt{3}-3}+\sqrt{3} \right ) }{36}}+{\frac{1}{3\,\sqrt{2\,\sqrt{3}+3}}\arctan \left ({\frac{2\,\tan \left ( x \right ) -\sqrt{2\,\sqrt{3}-3}}{\sqrt{2\,\sqrt{3}+3}}} \right ) }+{\frac{\sqrt{3}}{6\,\sqrt{2\,\sqrt{3}+3}}\arctan \left ({\frac{2\,\tan \left ( x \right ) -\sqrt{2\,\sqrt{3}-3}}{\sqrt{2\,\sqrt{3}+3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)^6),x)

[Out]

-1/3/tan(x)-1/36*3^(1/2)*(2*3^(1/2)-3)^(1/2)*ln(tan(x)^2+tan(x)*(2*3^(1/2)-3)^(1/2)+3^(1/2))+1/3/(2*3^(1/2)+3)
^(1/2)*arctan((2*tan(x)+(2*3^(1/2)-3)^(1/2))/(2*3^(1/2)+3)^(1/2))+1/6/(2*3^(1/2)+3)^(1/2)*arctan((2*tan(x)+(2*
3^(1/2)-3)^(1/2))/(2*3^(1/2)+3)^(1/2))*3^(1/2)+1/36*3^(1/2)*(2*3^(1/2)-3)^(1/2)*ln(tan(x)^2-tan(x)*(2*3^(1/2)-
3)^(1/2)+3^(1/2))+1/3/(2*3^(1/2)+3)^(1/2)*arctan((2*tan(x)-(2*3^(1/2)-3)^(1/2))/(2*3^(1/2)+3)^(1/2))+1/6/(2*3^
(1/2)+3)^(1/2)*arctan((2*tan(x)-(2*3^(1/2)-3)^(1/2))/(2*3^(1/2)+3)^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^6),x, algorithm="maxima")

[Out]

1/3*(3*(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)*integrate(1/3*((cos(3*x) + 4*cos(2*x) + cos(x))*cos(4*x) + (
14*cos(2*x) + 4*cos(x) + 1)*cos(3*x) + 2*cos(3*x)^2 + 2*(7*cos(x) + 2)*cos(2*x) + 24*cos(2*x)^2 + 2*cos(x)^2 +
(sin(3*x) + 4*sin(2*x) + sin(x))*sin(4*x) + 2*(7*sin(2*x) + 2*sin(x))*sin(3*x) + 2*sin(3*x)^2 + 24*sin(2*x)^2
+ 14*sin(2*x)*sin(x) + 2*sin(x)^2 + cos(x))/(2*(2*cos(3*x) + 6*cos(2*x) + 2*cos(x) + 1)*cos(4*x) + cos(4*x)^2
+ 4*(6*cos(2*x) + 2*cos(x) + 1)*cos(3*x) + 4*cos(3*x)^2 + 12*(2*cos(x) + 1)*cos(2*x) + 36*cos(2*x)^2 + 4*cos(
x)^2 + 4*(sin(3*x) + 3*sin(2*x) + sin(x))*sin(4*x) + sin(4*x)^2 + 8*(3*sin(2*x) + sin(x))*sin(3*x) + 4*sin(3*x
)^2 + 36*sin(2*x)^2 + 24*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1), x) - 3*(cos(2*x)^2 + sin(2*x)^2 - 2*cos
(2*x) + 1)*integrate(-1/3*((cos(3*x) - 4*cos(2*x) + cos(x))*cos(4*x) + (14*cos(2*x) - 4*cos(x) + 1)*cos(3*x) -
2*cos(3*x)^2 + 2*(7*cos(x) - 2)*cos(2*x) - 24*cos(2*x)^2 - 2*cos(x)^2 + (sin(3*x) - 4*sin(2*x) + sin(x))*sin(
4*x) + 2*(7*sin(2*x) - 2*sin(x))*sin(3*x) - 2*sin(3*x)^2 - 24*sin(2*x)^2 + 14*sin(2*x)*sin(x) - 2*sin(x)^2 + c
os(x))/(2*(2*cos(3*x) - 6*cos(2*x) + 2*cos(x) - 1)*cos(4*x) - cos(4*x)^2 + 4*(6*cos(2*x) - 2*cos(x) + 1)*cos(3
*x) - 4*cos(3*x)^2 + 12*(2*cos(x) - 1)*cos(2*x) - 36*cos(2*x)^2 - 4*cos(x)^2 + 4*(sin(3*x) - 3*sin(2*x) + sin(
x))*sin(4*x) - sin(4*x)^2 + 8*(3*sin(2*x) - sin(x))*sin(3*x) - 4*sin(3*x)^2 - 36*sin(2*x)^2 + 24*sin(2*x)*sin(
x) - 4*sin(x)^2 + 4*cos(x) - 1), x) - 2*sin(2*x))/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^6),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)**6),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{\cos \left (x\right )^{6} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^6),x, algorithm="giac")

[Out]

integrate(-1/(cos(x)^6 - 1), x)