\(\int \frac {1}{1-\cos ^6(x)} \, dx\) [84]

Optimal result

Integrand size = 10, antiderivative size = 71 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \]

[Out]

-1/3*cot(x)-1/3*arctan(cot(x)*(1+(-1)^(1/3))^(1/2))/(1+(-1)^(1/3))^(1/2)-1/3*arctan(cot(x)*(1-(-1)^(2/3))^(1/2
))/(1-(-1)^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3290, 3260, 209, 3254, 3852, 8} \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \]

[In]

Int[(1 - Cos[x]^6)^(-1),x]

[Out]

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

Rule 8

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

Rule 209

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

Rule 3254

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 3260

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 3290

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 3852

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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {Subst}\left (\int \frac {1}{1+\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {1}{3} \text {Subst}(\int 1 \, dx,x,\cot (x)) \\ & = -\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.65 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\frac {(15+8 \cos (2 x)+\cos (4 x)) \sin (x) \left (6 \cos (x)+i \sqrt [4]{-3} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (-i+\sqrt {3}\right ) \tan (x)\right ) \sin (x)+\sqrt [4]{-3} \left (-3 i+\sqrt {3}\right ) \arctan \left (\frac {(-1)^{3/4} \left (i+\sqrt {3}\right ) \tan (x)}{2 \sqrt [4]{3}}\right ) \sin (x)\right )}{144 \left (-1+\cos ^6(x)\right )} \]

[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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {2 i}{3 \left ({\mathrm e}^{2 i x}-1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3888 \textit {\_Z}^{4}+108 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+1296 i \textit {\_R}^{3}-216 \textit {\_R}^{2}-1\right )\right )\)	\(51\)
default	\(-\frac {\sqrt {3}\, \left (\frac {\sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\left (x \right )+\tan \left (x \right ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{2}+\frac {2 \left (-\sqrt {3}-\frac {3}{2}\right ) \arctan \left (\frac {2 \tan \left (x \right )+\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{\sqrt {2 \sqrt {3}+3}}\right )}{18}-\frac {\sqrt {3}\, \left (-\frac {\sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\left (x \right )-\tan \left (x \right ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{2}+\frac {2 \left (-\sqrt {3}-\frac {3}{2}\right ) \arctan \left (\frac {2 \tan \left (x \right )-\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{\sqrt {2 \sqrt {3}+3}}\right )}{18}-\frac {1}{3 \tan \left (x \right )}\)	\(173\)

[In]

int(1/(1-cos(x)^6),x,method=_RETURNVERBOSE)

[Out]

-2/3*I/(exp(2*I*x)-1)+sum(_R*ln(exp(2*I*x)+1296*I*_R^3-216*_R^2-1),_R=RootOf(3888*_Z^4+108*_Z^2+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.37 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\sqrt {6} \sqrt {i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (i \, \sqrt {3} - 3\right )}^{\frac {3}{2}} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} + 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} + 3\right ) \sin \left (x\right ) - \sqrt {6} \sqrt {i \, \sqrt {3} - 3} \log \left (\sqrt {6} \sqrt {i \, \sqrt {3} - 3} {\left (-i \, \sqrt {3} + 3\right )} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} + 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} + 3\right ) \sin \left (x\right ) + \sqrt {6} \sqrt {-i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (i \, \sqrt {3} + 3\right )} \sqrt {-i \, \sqrt {3} - 3} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} - 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} - 3\right ) \sin \left (x\right ) - \sqrt {6} \sqrt {-i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {3}{2}} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} - 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} - 3\right ) \sin \left (x\right ) + 24 \, \cos \left (x\right )}{72 \, \sin \left (x\right )} \]

[In]

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

[Out]

-1/72*(sqrt(6)*sqrt(I*sqrt(3) - 3)*log(sqrt(6)*(I*sqrt(3) - 3)^(3/2)*cos(x)*sin(x) - 6*(-I*sqrt(3) + 2)*cos(x)
^2 - 3*I*sqrt(3) + 3)*sin(x) - sqrt(6)*sqrt(I*sqrt(3) - 3)*log(sqrt(6)*sqrt(I*sqrt(3) - 3)*(-I*sqrt(3) + 3)*co
s(x)*sin(x) - 6*(-I*sqrt(3) + 2)*cos(x)^2 - 3*I*sqrt(3) + 3)*sin(x) + sqrt(6)*sqrt(-I*sqrt(3) - 3)*log(sqrt(6)
*(I*sqrt(3) + 3)*sqrt(-I*sqrt(3) - 3)*cos(x)*sin(x) - 6*(-I*sqrt(3) - 2)*cos(x)^2 - 3*I*sqrt(3) - 3)*sin(x) -
sqrt(6)*sqrt(-I*sqrt(3) - 3)*log(sqrt(6)*(-I*sqrt(3) - 3)^(3/2)*cos(x)*sin(x) - 6*(-I*sqrt(3) - 2)*cos(x)^2 -
3*I*sqrt(3) - 3)*sin(x) + 24*cos(x))/sin(x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (66) = 132\).

Time = 8.91 (sec) , antiderivative size = 728, normalized size of antiderivative = 10.25 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

sqrt(2)*3**(3/4)*(atan(sqrt(2)*3**(1/4)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/pi))/36 + sqrt(2)*3**(1/4)*(atan
(sqrt(2)*3**(1/4)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/pi))/12 + sqrt(2)*3**(3/4)*(atan(sqrt(2)*3**(1/4)*tan(
x/2) + 1) + pi*floor((x/2 - pi/2)/pi))/36 + sqrt(2)*3**(1/4)*(atan(sqrt(2)*3**(1/4)*tan(x/2) + 1) + pi*floor((
x/2 - pi/2)/pi))/12 + sqrt(2)*3**(3/4)*(atan(sqrt(2)*3**(3/4)*tan(x/2)/3 - 1) + pi*floor((x/2 - pi/2)/pi))/36
+ sqrt(2)*3**(1/4)*(atan(sqrt(2)*3**(3/4)*tan(x/2)/3 - 1) + pi*floor((x/2 - pi/2)/pi))/12 + sqrt(2)*3**(3/4)*(
atan(sqrt(2)*3**(3/4)*tan(x/2)/3 + 1) + pi*floor((x/2 - pi/2)/pi))/36 + sqrt(2)*3**(1/4)*(atan(sqrt(2)*3**(3/4
)*tan(x/2)/3 + 1) + pi*floor((x/2 - pi/2)/pi))/12 - sqrt(2)*3**(1/4)*log(4*tan(x/2)**2 - 4*sqrt(2)*3**(1/4)*ta
n(x/2) + 4*sqrt(3))/24 + sqrt(2)*3**(3/4)*log(4*tan(x/2)**2 - 4*sqrt(2)*3**(1/4)*tan(x/2) + 4*sqrt(3))/72 - sq
rt(2)*3**(3/4)*log(4*tan(x/2)**2 + 4*sqrt(2)*3**(1/4)*tan(x/2) + 4*sqrt(3))/72 + sqrt(2)*3**(1/4)*log(4*tan(x/
2)**2 + 4*sqrt(2)*3**(1/4)*tan(x/2) + 4*sqrt(3))/24 - sqrt(2)*3**(3/4)*log(36*tan(x/2)**2 - 12*sqrt(2)*3**(3/4
)*tan(x/2) + 12*sqrt(3))/72 + sqrt(2)*3**(1/4)*log(36*tan(x/2)**2 - 12*sqrt(2)*3**(3/4)*tan(x/2) + 12*sqrt(3))
/24 - sqrt(2)*3**(1/4)*log(36*tan(x/2)**2 + 12*sqrt(2)*3**(3/4)*tan(x/2) + 12*sqrt(3))/24 + sqrt(2)*3**(3/4)*l
og(36*tan(x/2)**2 + 12*sqrt(2)*3**(3/4)*tan(x/2) + 12*sqrt(3))/72 + tan(x/2)/6 - 1/(6*tan(x/2))

Maxima [F]

\[ \int \frac {1}{1-\cos ^6(x)} \, dx=\int { -\frac {1}{\cos \left (x\right )^{6} - 1} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (49) = 98\).

Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.80 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} + 4 \, \tan \left (x\right )\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} + \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} - 4 \, \tan \left (x\right )\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} - \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {3}\right ) + \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {3}\right ) - \frac {1}{3 \, \tan \left (x\right )} \]

[In]

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

[Out]

1/18*(pi*floor(x/pi + 1/2) - arctan(-1/3*3^(3/4)*(3^(1/4)*(sqrt(6) - sqrt(2)) + 4*tan(x))/(sqrt(6) + sqrt(2)))
)*sqrt(6*sqrt(3) + 9) + 1/18*(pi*floor(x/pi + 1/2) + arctan(-1/3*3^(3/4)*(3^(1/4)*(sqrt(6) - sqrt(2)) - 4*tan(
x))/(sqrt(6) + sqrt(2))))*sqrt(6*sqrt(3) + 9) - 1/36*sqrt(6*sqrt(3) - 9)*log(1/2*(sqrt(6)*3^(1/4) - 3^(1/4)*sq
rt(2))*tan(x) + tan(x)^2 + sqrt(3)) + 1/36*sqrt(6*sqrt(3) - 9)*log(-1/2*(sqrt(6)*3^(1/4) - 3^(1/4)*sqrt(2))*ta
n(x) + tan(x)^2 + sqrt(3)) - 1/3/tan(x)

Mupad [B] (verification not implemented)

Time = 3.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.34 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {1}{3\,\mathrm {tan}\left (x\right )}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\left (x\right )\,\left (\frac {1}{27}-\frac {1}{27}{}\mathrm {i}\right )}{-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1+1{}\mathrm {i}\right )+3^{3/4}\,\left (-1+1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\left (x\right )\,\left (\frac {1}{27}+\frac {1}{27}{}\mathrm {i}\right )}{\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1-\mathrm {i}\right )+3^{3/4}\,\left (-1-\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36} \]

[In]

int(-1/(cos(x)^6 - 1),x)

[Out]

(6^(1/2)*atan((3^(1/4)*6^(1/2)*tan(x)*(1/27 - 1i/27))/((3^(1/2)*1i)/9 - 1/9))*(3^(1/4)*(1 + 1i) - 3^(3/4)*(1 -
1i))*1i)/36 - 1/(3*tan(x)) + (6^(1/2)*atan((3^(1/4)*6^(1/2)*tan(x)*(1/27 + 1i/27))/((3^(1/2)*1i)/9 + 1/9))*(3
^(1/4)*(1 - 1i) - 3^(3/4)*(1 + 1i))*1i)/36