3.83 \(\int \frac {1}{1-\cos ^5(x)} \, dx\)

Optimal. Leaf size=205 \[ \frac {2 \tan ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {\sin (x)}{5 (1-\cos (x))}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {(-1)^{4/5}-1}}+\frac {2 \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}} \]

[Out]

-1/5*sin(x)/(1-cos(x))+2/5*arctan(((1-(-1)^(3/5))/(1+(-1)^(3/5)))^(1/2)*tan(1/2*x))/(1+(-1)^(1/5))^(1/2)+2/5*a
rctan(((1-(-1)^(1/5))/(1+(-1)^(1/5)))^(1/2)*tan(1/2*x))/(1-(-1)^(2/5))^(1/2)+2/5*arctanh(((-1-(-1)^(4/5))/(1-(
-1)^(4/5)))^(1/2)*tan(1/2*x))/(-1-(-1)^(3/5))^(1/2)-2/5*arctanh(tan(1/2*x)/((-1+(-1)^(2/5))/(1+(-1)^(2/5)))^(1
/2))/(-1+(-1)^(4/5))^(1/2)

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Rubi [A]  time = 0.47, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3213, 2648, 2659, 205, 208} \[ \frac {2 \tan ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {\sin (x)}{5 (1-\cos (x))}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {(-1)^{4/5}-1}}+\frac {2 \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x]^5)^(-1),x]

[Out]

(2*ArcTan[Sqrt[(1 - (-1)^(1/5))/(1 + (-1)^(1/5))]*Tan[x/2]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTan[Sqrt[(1 - (-
1)^(3/5))/(1 + (-1)^(3/5))]*Tan[x/2]])/(5*Sqrt[1 + (-1)^(1/5)]) - (2*ArcTanh[Tan[x/2]/Sqrt[-((1 - (-1)^(2/5))/
(1 + (-1)^(2/5)))]])/(5*Sqrt[-1 + (-1)^(4/5)]) + (2*ArcTanh[Sqrt[-((1 + (-1)^(4/5))/(1 - (-1)^(4/5)))]*Tan[x/2
]])/(5*Sqrt[-1 - (-1)^(3/5)]) - Sin[x]/(5*(1 - Cos[x]))

Rule 205

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

Rule 208

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3213

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

\begin {align*} \int \frac {1}{1-\cos ^5(x)} \, dx &=\int \left (\frac {1}{5 (1-\cos (x))}+\frac {1}{5 \left (1+\sqrt [5]{-1} \cos (x)\right )}+\frac {1}{5 \left (1-(-1)^{2/5} \cos (x)\right )}+\frac {1}{5 \left (1+(-1)^{3/5} \cos (x)\right )}+\frac {1}{5 \left (1-(-1)^{4/5} \cos (x)\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{1-\cos (x)} \, dx+\frac {1}{5} \int \frac {1}{1+\sqrt [5]{-1} \cos (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{2/5} \cos (x)} \, dx+\frac {1}{5} \int \frac {1}{1+(-1)^{3/5} \cos (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{4/5} \cos (x)} \, dx\\ &=-\frac {\sin (x)}{5 (1-\cos (x))}+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [5]{-1}+\left (1-\sqrt [5]{-1}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1-(-1)^{2/5}+\left (1+(-1)^{2/5}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1+(-1)^{3/5}+\left (1-(-1)^{3/5}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1-(-1)^{4/5}+\left (1+(-1)^{4/5}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {-1+(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}-\frac {\sin (x)}{5 (1-\cos (x))}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 378, normalized size = 1.84 \[ -\frac {1}{5} \cot \left (\frac {x}{2}\right )+\frac {1}{10} \text {RootSum}\left [\text {$\#$1}^8+2 \text {$\#$1}^7+8 \text {$\#$1}^6+14 \text {$\#$1}^5+30 \text {$\#$1}^4+14 \text {$\#$1}^3+8 \text {$\#$1}^2+2 \text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^6 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+8 \text {$\#$1}^5 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+30 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+80 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-15 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-4 i \text {$\#$1} \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+30 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-4 i \text {$\#$1}^5 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-15 i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-40 i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+8 \text {$\#$1} \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )}{4 \text {$\#$1}^7+7 \text {$\#$1}^6+24 \text {$\#$1}^5+35 \text {$\#$1}^4+60 \text {$\#$1}^3+21 \text {$\#$1}^2+8 \text {$\#$1}+1}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cos[x]^5)^(-1),x]

[Out]

-1/5*Cot[x/2] + RootSum[1 + 2*#1 + 8*#1^2 + 14*#1^3 + 30*#1^4 + 14*#1^5 + 8*#1^6 + 2*#1^7 + #1^8 & , (2*ArcTan
[Sin[x]/(Cos[x] - #1)] - I*Log[1 - 2*Cos[x]*#1 + #1^2] + 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - (4*I)*Log[1 - 2*C
os[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + 80*ArcT
an[Sin[x]/(Cos[x] - #1)]*#1^3 - (40*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4
- (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 + 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 - (4*I)*Log[1 - 2*Cos[x]*#1 +
#1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(1 + 8*#1 + 21*#1^2 +
60*#1^3 + 35*#1^4 + 24*#1^5 + 7*#1^6 + 4*#1^7) & ]/10

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^5),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^5),x, algorithm="giac")

[Out]

sage0*x

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maple [C]  time = 0.08, size = 62, normalized size = 0.30 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+10 \textit {\_Z}^{4}+5\right )}{\sum }\frac {\left (\textit {\_R}^{6}+5 \textit {\_R}^{4}+5 \textit {\_R}^{2}+5\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+5 \textit {\_R}^{3}}\right )}{10}-\frac {1}{5 \tan \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)^5),x)

[Out]

1/10*sum((_R^6+5*_R^4+5*_R^2+5)/(_R^7+5*_R^3)*ln(tan(1/2*x)-_R),_R=RootOf(_Z^8+10*_Z^4+5))-1/5/tan(1/2*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)^5),x, algorithm="maxima")

[Out]

1/5*(5*(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)*integrate(2/5*((cos(7*x) + 4*cos(6*x) + 15*cos(5*x) + 40*cos(4*x)
+ 15*cos(3*x) + 4*cos(2*x) + cos(x))*cos(8*x) + (16*cos(6*x) + 44*cos(5*x) + 110*cos(4*x) + 44*cos(3*x) + 16*c
os(2*x) + 4*cos(x) + 1)*cos(7*x) + 2*cos(7*x)^2 + 4*(44*cos(5*x) + 110*cos(4*x) + 44*cos(3*x) + 16*cos(2*x) +
4*cos(x) + 1)*cos(6*x) + 32*cos(6*x)^2 + (1010*cos(4*x) + 420*cos(3*x) + 176*cos(2*x) + 44*cos(x) + 15)*cos(5*
x) + 210*cos(5*x)^2 + 10*(101*cos(3*x) + 44*cos(2*x) + 11*cos(x) + 4)*cos(4*x) + 1200*cos(4*x)^2 + (176*cos(2*
x) + 44*cos(x) + 15)*cos(3*x) + 210*cos(3*x)^2 + 4*(4*cos(x) + 1)*cos(2*x) + 32*cos(2*x)^2 + 2*cos(x)^2 + (sin
(7*x) + 4*sin(6*x) + 15*sin(5*x) + 40*sin(4*x) + 15*sin(3*x) + 4*sin(2*x) + sin(x))*sin(8*x) + 2*(8*sin(6*x) +
22*sin(5*x) + 55*sin(4*x) + 22*sin(3*x) + 8*sin(2*x) + 2*sin(x))*sin(7*x) + 2*sin(7*x)^2 + 8*(22*sin(5*x) + 5
5*sin(4*x) + 22*sin(3*x) + 8*sin(2*x) + 2*sin(x))*sin(6*x) + 32*sin(6*x)^2 + 2*(505*sin(4*x) + 210*sin(3*x) +
88*sin(2*x) + 22*sin(x))*sin(5*x) + 210*sin(5*x)^2 + 10*(101*sin(3*x) + 44*sin(2*x) + 11*sin(x))*sin(4*x) + 12
00*sin(4*x)^2 + 44*(4*sin(2*x) + sin(x))*sin(3*x) + 210*sin(3*x)^2 + 32*sin(2*x)^2 + 16*sin(2*x)*sin(x) + 2*si
n(x)^2 + cos(x))/(2*(2*cos(7*x) + 8*cos(6*x) + 14*cos(5*x) + 30*cos(4*x) + 14*cos(3*x) + 8*cos(2*x) + 2*cos(x)
+ 1)*cos(8*x) + cos(8*x)^2 + 4*(8*cos(6*x) + 14*cos(5*x) + 30*cos(4*x) + 14*cos(3*x) + 8*cos(2*x) + 2*cos(x)
+ 1)*cos(7*x) + 4*cos(7*x)^2 + 16*(14*cos(5*x) + 30*cos(4*x) + 14*cos(3*x) + 8*cos(2*x) + 2*cos(x) + 1)*cos(6*
x) + 64*cos(6*x)^2 + 28*(30*cos(4*x) + 14*cos(3*x) + 8*cos(2*x) + 2*cos(x) + 1)*cos(5*x) + 196*cos(5*x)^2 + 60
*(14*cos(3*x) + 8*cos(2*x) + 2*cos(x) + 1)*cos(4*x) + 900*cos(4*x)^2 + 28*(8*cos(2*x) + 2*cos(x) + 1)*cos(3*x)
+ 196*cos(3*x)^2 + 16*(2*cos(x) + 1)*cos(2*x) + 64*cos(2*x)^2 + 4*cos(x)^2 + 4*(sin(7*x) + 4*sin(6*x) + 7*sin
(5*x) + 15*sin(4*x) + 7*sin(3*x) + 4*sin(2*x) + sin(x))*sin(8*x) + sin(8*x)^2 + 8*(4*sin(6*x) + 7*sin(5*x) + 1
5*sin(4*x) + 7*sin(3*x) + 4*sin(2*x) + sin(x))*sin(7*x) + 4*sin(7*x)^2 + 32*(7*sin(5*x) + 15*sin(4*x) + 7*sin(
3*x) + 4*sin(2*x) + sin(x))*sin(6*x) + 64*sin(6*x)^2 + 56*(15*sin(4*x) + 7*sin(3*x) + 4*sin(2*x) + sin(x))*sin
(5*x) + 196*sin(5*x)^2 + 120*(7*sin(3*x) + 4*sin(2*x) + sin(x))*sin(4*x) + 900*sin(4*x)^2 + 56*(4*sin(2*x) + s
in(x))*sin(3*x) + 196*sin(3*x)^2 + 64*sin(2*x)^2 + 32*sin(2*x)*sin(x) + 4*sin(x)^2 + 4*cos(x) + 1), x) - 2*sin
(x))/(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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mupad [B]  time = 2.45, size = 403, normalized size = 1.97 \[ 2\,\mathrm {atanh}\left (\frac {50\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-20\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{5\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}+2\,\sqrt {5}-10\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}-5}\right )\,\sqrt {\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-2\,\mathrm {atanh}\left (\frac {50\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-20\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{5\,\sqrt {5}\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}-2\,\sqrt {5}-10\,\sqrt {-\frac {2\,\sqrt {5}}{5}-1}+5}\right )\,\sqrt {-\frac {\sqrt {-\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-\frac {\mathrm {cot}\left (\frac {x}{2}\right )}{5}+2\,\mathrm {atanh}\left (\frac {50\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}+20\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{5\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}-2\,\sqrt {5}+10\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}-5}\right )\,\sqrt {-\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}-2\,\mathrm {atanh}\left (\frac {50\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}+20\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}}}{5\,\sqrt {5}\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}+2\,\sqrt {5}+10\,\sqrt {\frac {2\,\sqrt {5}}{5}-1}+5}\right )\,\sqrt {\frac {\sqrt {\frac {2\,\sqrt {5}}{5}-1}}{50}-\frac {1}{50}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cos(x)^5 - 1),x)

[Out]

2*atanh((50*tan(x/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 20*5^(1/2)*tan(x/2)*((- (2*5^(1/2))/5 - 1
)^(1/2)/50 - 1/50)^(1/2))/(5*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2) + 2*5^(1/2) - 10*(- (2*5^(1/2))/5 - 1)^(1/2)
- 5))*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 2*atanh((50*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 -
1/50)^(1/2) - 20*5^(1/2)*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*5^(1/2)*(- (2*5^(1/2))/
5 - 1)^(1/2) - 2*5^(1/2) - 10*(- (2*5^(1/2))/5 - 1)^(1/2) + 5))*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2
) - cot(x/2)/5 + 2*atanh((50*tan(x/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) + 20*5^(1/2)*tan(x/2)*(- (
(2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2) - 2*5^(1/2) + 10*((2*5^(1/2))/
5 - 1)^(1/2) - 5))*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 2*atanh((50*tan(x/2)*(((2*5^(1/2))/5 - 1)^(
1/2)/50 - 1/50)^(1/2) + 20*5^(1/2)*tan(x/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*5^(1/2)*((2*5^(1/2
))/5 - 1)^(1/2) + 2*5^(1/2) + 10*((2*5^(1/2))/5 - 1)^(1/2) + 5))*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)**5),x)

[Out]

Timed out

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