3.80 \(\int \frac {1}{1+\cos ^5(x)} \, dx\)
Optimal. Leaf size=223 \[ \frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {\sin (x)}{5 (\cos (x)+1)}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {(-1)^{2/5}-1}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )} \]
[Out]
1/5*sin(x)/(1+cos(x))-2/5*arctanh(tan(1/2*x)/((-1+(-1)^(1/5))/(1+(-1)^(1/5)))^(1/2))/(-1+(-1)^(2/5))^(1/2)+2/5
*arctan(((1-(-1)^(4/5))/(1+(-1)^(4/5)))^(1/2)*tan(1/2*x))/(1+(-1)^(3/5))^(1/2)-2/5*arctanh(((-1-(-1)^(3/5))/(1
-(-1)^(3/5)))^(1/2)*tan(1/2*x))*((-1-(-1)^(3/5))/(1-(-1)^(3/5)))^(1/2)/(1+(-1)^(3/5))+2/5*arctan(((1-(-1)^(2/5
))/(1+(-1)^(2/5)))^(1/2)*tan(1/2*x))/(1-(-1)^(4/5))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.56, antiderivative size = 223, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used =
{3213, 2648, 2659, 208, 205} \[ \frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {\sin (x)}{5 (\cos (x)+1)}-\frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {(-1)^{2/5}-1}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Cos[x]^5)^(-1),x]
[Out]
(2*ArcTan[Sqrt[(1 - (-1)^(2/5))/(1 + (-1)^(2/5))]*Tan[x/2]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTan[Sqrt[(1 - (-
1)^(4/5))/(1 + (-1)^(4/5))]*Tan[x/2]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTanh[Tan[x/2]/Sqrt[-((1 - (-1)^(1/5))/
(1 + (-1)^(1/5)))]])/(5*Sqrt[-1 + (-1)^(2/5)]) - (2*Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*ArcTanh[Sqrt[-(
(1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*Tan[x/2]])/(5*(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\\ &=-\left (\frac {1}{5} \int \frac {1}{-1-\cos (x)} \, dx\right )-\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-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-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 {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {-1+(-1)^{2/5}}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {\sin (x)}{5 (1+\cos (x))}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 378, normalized size = 1.70 \[ \frac {1}{5} \tan \left (\frac {x}{2}\right )-\frac {1}{10} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^7+8 \text {$\#$1}^6-14 \text {$\#$1}^5+30 \text {$\#$1}^4-14 \text {$\#$1}^3+8 \text {$\#$1}^2-2 \text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^6 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-8 \text {$\#$1}^5 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+30 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-80 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-15 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+4 i \text {$\#$1} \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+30 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+4 i \text {$\#$1}^5 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-15 i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+40 i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-8 \text {$\#$1} \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )}{4 \text {$\#$1}^7-7 \text {$\#$1}^6+24 \text {$\#$1}^5-35 \text {$\#$1}^4+60 \text {$\#$1}^3-21 \text {$\#$1}^2+8 \text {$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + Cos[x]^5)^(-1),x]
[Out]
-1/10*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]/(C
os[x] - #1)] - I*Log[1 - 2*Cos[x]*#1 + #1^2] - 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 + (4*I)*Log[1 - 2*Cos[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*ArcTan[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) & ] + Tan[x/2]/5
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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 = 0.07, size = 62, normalized size = 0.28 \[ \frac {\tan \left (\frac {x}{2}\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (5 \textit {\_Z}^{8}+10 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (5 \textit {\_R}^{6}+5 \textit {\_R}^{4}+5 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+\textit {\_R}^{3}}\right )}{50} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+cos(x)^5),x)
[Out]
1/5*tan(1/2*x)+1/50*sum((5*_R^6+5*_R^4+5*_R^2+1)/(_R^7+_R^3)*ln(tan(1/2*x)-_R),_R=RootOf(5*_Z^8+10*_Z^4+1))
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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
*cos(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 + (s
in(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) -
55*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) -
1200*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*
sin(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*s
in(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) +
15*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*si
n(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))*s
in(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) -
sin(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*s
in(x))/(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)
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mupad [B] time = 2.78, size = 535, normalized size = 2.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cos(x)^5 + 1),x)
[Out]
tan(x/2)/5 + 2*atanh((603979776*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(244140625*((3355443
2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 - (134217728*5^(1/2))/1220703125 + (67108864*(- (2*5^(1/2))/
5 - 1)^(1/2))/1220703125 - 301989888/1220703125)) + (268435456*5^(1/2)*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)
/50 - 1/50)^(1/2))/(244140625*((33554432*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 - (134217728*5^(1/2))
/1220703125 + (67108864*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 - 301989888/1220703125)))*(- (- (2*5^(1/2))/5
- 1)^(1/2)/50 - 1/50)^(1/2) - 2*atanh((603979776*tan(x/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(2441
40625*((33554432*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 + (134217728*5^(1/2))/1220703125 + (67108864*
(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 + 301989888/1220703125)) + (268435456*5^(1/2)*tan(x/2)*((- (2*5^(1/2))
/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(244140625*((33554432*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 + (13421
7728*5^(1/2))/1220703125 + (67108864*(- (2*5^(1/2))/5 - 1)^(1/2))/1220703125 + 301989888/1220703125)))*((- (2*
5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 2*atanh((603979776*tan(x/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(
1/2))/(244140625*((33554432*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 - (134217728*5^(1/2))/1220703125 - (
67108864*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 + 301989888/1220703125)) - (268435456*5^(1/2)*tan(x/2)*(- ((2*5
^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(244140625*((33554432*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 - (
134217728*5^(1/2))/1220703125 - (67108864*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 + 301989888/1220703125)))*(- (
(2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) + 2*atanh((603979776*tan(x/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^
(1/2))/(244140625*((33554432*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 + (134217728*5^(1/2))/1220703125 -
(67108864*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 - 301989888/1220703125)) - (268435456*5^(1/2)*tan(x/2)*(((2*5^
(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(244140625*((33554432*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 + (1
34217728*5^(1/2))/1220703125 - (67108864*((2*5^(1/2))/5 - 1)^(1/2))/1220703125 - 301989888/1220703125)))*(((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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