3.3.58 \(\int \frac {1}{1-\sin ^5(x)} \, dx\) [258]
Optimal. Leaf size=187 \[ -\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}+\frac {\cos (x)}{5 (1-\sin (x))} \]
[Out]
1/5*cos(x)/(1-sin(x))+2/5*arctan(((-1)^(3/5)+tan(1/2*x))/(1+(-1)^(1/5))^(1/2))/(1+(-1)^(1/5))^(1/2)+2/5*arctan
(((-1)^(1/5)+tan(1/2*x))/(1-(-1)^(2/5))^(1/2))/(1-(-1)^(2/5))^(1/2)-2/5*arctan(((-1)^(4/5)-tan(1/2*x))/(1+(-1)
^(3/5))^(1/2))/(1+(-1)^(3/5))^(1/2)-2/5*arctan(((-1)^(2/5)-tan(1/2*x))/(1-(-1)^(4/5))^(1/2))/(1-(-1)^(4/5))^(1
/2)
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Rubi [A]
time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3292, 2727,
2739, 632, 210} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {(-1)^{2/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}-\frac {2 \text {ArcTan}\left (\frac {(-1)^{4/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {2 \text {ArcTan}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt [5]{-1}}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \text {ArcTan}\left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{3/5}}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}+\frac {\cos (x)}{5 (1-\sin (x))} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(1 - Sin[x]^5)^(-1),x]
[Out]
(-2*ArcTan[((-1)^(2/5) - Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) - (2*ArcTan[((-1)^(4/5) - T
an[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) + (2*ArcTan[((-1)^(1/5) + Tan[x/2])/Sqrt[1 - (-1)^(2/
5)]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTan[((-1)^(3/5) + Tan[x/2])/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/
5)]) + Cos[x]/(5*(1 - Sin[x]))
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 2727
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 2739
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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 3292
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-\sin ^5(x)} \, dx &=\int \left (\frac {1}{5 (1-\sin (x))}+\frac {1}{5 \left (1+\sqrt [5]{-1} \sin (x)\right )}+\frac {1}{5 \left (1-(-1)^{2/5} \sin (x)\right )}+\frac {1}{5 \left (1+(-1)^{3/5} \sin (x)\right )}+\frac {1}{5 \left (1-(-1)^{4/5} \sin (x)\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{1-\sin (x)} \, dx+\frac {1}{5} \int \frac {1}{1+\sqrt [5]{-1} \sin (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{2/5} \sin (x)} \, dx+\frac {1}{5} \int \frac {1}{1+(-1)^{3/5} \sin (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{4/5} \sin (x)} \, dx\\ &=\frac {\cos (x)}{5 (1-\sin (x))}+\frac {2}{5} \text {Subst}\left (\int \frac {1}{1+2 \sqrt [5]{-1} x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \text {Subst}\left (\int \frac {1}{1-2 (-1)^{2/5} x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \text {Subst}\left (\int \frac {1}{1+2 (-1)^{3/5} x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {2}{5} \text {Subst}\left (\int \frac {1}{1-2 (-1)^{4/5} x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {\cos (x)}{5 (1-\sin (x))}-\frac {4}{5} \text {Subst}\left (\int \frac {1}{-4 \left (1+\sqrt [5]{-1}\right )-x^2} \, dx,x,2 (-1)^{3/5}+2 \tan \left (\frac {x}{2}\right )\right )-\frac {4}{5} \text {Subst}\left (\int \frac {1}{-4 \left (1-(-1)^{2/5}\right )-x^2} \, dx,x,2 \sqrt [5]{-1}+2 \tan \left (\frac {x}{2}\right )\right )-\frac {4}{5} \text {Subst}\left (\int \frac {1}{-4 \left (1+(-1)^{3/5}\right )-x^2} \, dx,x,-2 (-1)^{4/5}+2 \tan \left (\frac {x}{2}\right )\right )-\frac {4}{5} \text {Subst}\left (\int \frac {1}{-4 \left (1-(-1)^{4/5}\right )-x^2} \, dx,x,-2 (-1)^{2/5}+2 \tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}+\frac {\cos (x)}{5 (1-\sin (x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.10, size = 413, normalized size = 2.21 \begin {gather*} \frac {1}{10} i \text {RootSum}\left [1-2 i \text {$\#$1}-8 \text {$\#$1}^2+14 i \text {$\#$1}^3+30 \text {$\#$1}^4-14 i \text {$\#$1}^5-8 \text {$\#$1}^6+2 i \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )+8 i \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}+4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+30 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-80 i \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-40 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-30 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4+15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+8 i \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^5+4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-i-8 \text {$\#$1}+21 i \text {$\#$1}^2+60 \text {$\#$1}^3-35 i \text {$\#$1}^4-24 \text {$\#$1}^5+7 i \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {2 \sin \left (\frac {x}{2}\right )}{5 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 - Sin[x]^5)^(-1),x]
[Out]
(I/10)*RootSum[1 - (2*I)*#1 - 8*#1^2 + (14*I)*#1^3 + 30*#1^4 - (14*I)*#1^5 - 8*#1^6 + (2*I)*#1^7 + #1^8 & , (-
2*ArcTan[Sin[x]/(Cos[x] - #1)] + I*Log[1 - 2*Cos[x]*#1 + #1^2] + (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 + 4*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*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - 40*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*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 + 4*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)/(-I - 8*#1 + (
21*I)*#1^2 + 60*#1^3 - (35*I)*#1^4 - 24*#1^5 + (7*I)*#1^6 + 4*#1^7) & ] + (2*Sin[x/2])/(5*(Cos[x/2] - Sin[x/2]
))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.66, size = 133, normalized size = 0.71
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {2}{5 \left ({\mathrm e}^{i x}-i\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (1953125 \textit {\_Z}^{8}+156250 \textit {\_Z}^{6}+6250 \textit {\_Z}^{4}+125 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-2343750 \textit {\_R}^{7}-234375 i \textit {\_R}^{6}-140625 \textit {\_R}^{5}-15625 i \textit {\_R}^{4}-4375 \textit {\_R}^{3}-500 i \textit {\_R}^{2}-50 \textit {\_R} -6 i\right )\right )\) |
\(87\) |
| default |
\(\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+2 \textit {\_Z}^{7}+8 \textit {\_Z}^{6}+14 \textit {\_Z}^{5}+30 \textit {\_Z}^{4}+14 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}+3 \textit {\_R}^{5}+10 \textit {\_R}^{4}+10 \textit {\_R}^{3}+10 \textit {\_R}^{2}+3 \textit {\_R} +2\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}+7 \textit {\_R}^{6}+24 \textit {\_R}^{5}+35 \textit {\_R}^{4}+60 \textit {\_R}^{3}+21 \textit {\_R}^{2}+8 \textit {\_R} +1}\right )}{5}-\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )-1\right )}\) |
\(133\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1-sin(x)^5),x,method=_RETURNVERBOSE)
[Out]
2/5*sum((2*_R^6+3*_R^5+10*_R^4+10*_R^3+10*_R^2+3*_R+2)/(4*_R^7+7*_R^6+24*_R^5+35*_R^4+60*_R^3+21*_R^2+8*_R+1)*
ln(tan(1/2*x)-_R),_R=RootOf(_Z^8+2*_Z^7+8*_Z^6+14*_Z^5+30*_Z^4+14*_Z^3+8*_Z^2+2*_Z+1))-2/5/(tan(1/2*x)-1)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-sin(x)^5),x, algorithm="maxima")
[Out]
1/5*(5*(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)*integrate(2/5*((4*cos(6*x) - 40*cos(4*x) + 4*cos(2*x) + sin(7*x) -
15*sin(5*x) + 15*sin(3*x) - sin(x))*cos(8*x) + 2*(22*cos(5*x) - 22*cos(3*x) + 2*cos(x) + 8*sin(6*x) - 55*sin(
4*x) + 8*sin(2*x))*cos(7*x) - 2*cos(7*x)^2 + 4*(110*cos(4*x) - 16*cos(2*x) + 44*sin(5*x) - 44*sin(3*x) + 4*sin
(x) + 1)*cos(6*x) - 32*cos(6*x)^2 + 2*(210*cos(3*x) - 22*cos(x) + 505*sin(4*x) - 88*sin(2*x))*cos(5*x) - 210*c
os(5*x)^2 + 10*(44*cos(2*x) + 101*sin(3*x) - 11*sin(x) - 4)*cos(4*x) - 1200*cos(4*x)^2 + 44*(cos(x) + 4*sin(2*
x))*cos(3*x) - 210*cos(3*x)^2 + 4*(4*sin(x) + 1)*cos(2*x) - 32*cos(2*x)^2 - 2*cos(x)^2 - (cos(7*x) - 15*cos(5*
x) + 15*cos(3*x) - cos(x) - 4*sin(6*x) + 40*sin(4*x) - 4*sin(2*x))*sin(8*x) - (16*cos(6*x) - 110*cos(4*x) + 16
*cos(2*x) - 44*sin(5*x) + 44*sin(3*x) - 4*sin(x) - 1)*sin(7*x) - 2*sin(7*x)^2 - 8*(22*cos(5*x) - 22*cos(3*x) +
2*cos(x) - 55*sin(4*x) + 8*sin(2*x))*sin(6*x) - 32*sin(6*x)^2 - (1010*cos(4*x) - 176*cos(2*x) - 420*sin(3*x)
+ 44*sin(x) + 15)*sin(5*x) - 210*sin(5*x)^2 - 10*(101*cos(3*x) - 11*cos(x) - 44*sin(2*x))*sin(4*x) - 1200*sin(
4*x)^2 - (176*cos(2*x) - 44*sin(x) - 15)*sin(3*x) - 210*sin(3*x)^2 - 16*cos(x)*sin(2*x) - 32*sin(2*x)^2 - 2*si
n(x)^2 - sin(x))/(2*(8*cos(6*x) - 30*cos(4*x) + 8*cos(2*x) + 2*sin(7*x) - 14*sin(5*x) + 14*sin(3*x) - 2*sin(x)
- 1)*cos(8*x) - cos(8*x)^2 + 8*(7*cos(5*x) - 7*cos(3*x) + cos(x) + 4*sin(6*x) - 15*sin(4*x) + 4*sin(2*x))*cos
(7*x) - 4*cos(7*x)^2 + 16*(30*cos(4*x) - 8*cos(2*x) + 14*sin(5*x) - 14*sin(3*x) + 2*sin(x) + 1)*cos(6*x) - 64*
cos(6*x)^2 + 56*(7*cos(3*x) - cos(x) + 15*sin(4*x) - 4*sin(2*x))*cos(5*x) - 196*cos(5*x)^2 + 60*(8*cos(2*x) +
14*sin(3*x) - 2*sin(x) - 1)*cos(4*x) - 900*cos(4*x)^2 + 56*(cos(x) + 4*sin(2*x))*cos(3*x) - 196*cos(3*x)^2 + 1
6*(2*sin(x) + 1)*cos(2*x) - 64*cos(2*x)^2 - 4*cos(x)^2 - 4*(cos(7*x) - 7*cos(5*x) + 7*cos(3*x) - cos(x) - 4*si
n(6*x) + 15*sin(4*x) - 4*sin(2*x))*sin(8*x) - sin(8*x)^2 - 4*(8*cos(6*x) - 30*cos(4*x) + 8*cos(2*x) - 14*sin(5
*x) + 14*sin(3*x) - 2*sin(x) - 1)*sin(7*x) - 4*sin(7*x)^2 - 32*(7*cos(5*x) - 7*cos(3*x) + cos(x) - 15*sin(4*x)
+ 4*sin(2*x))*sin(6*x) - 64*sin(6*x)^2 - 28*(30*cos(4*x) - 8*cos(2*x) - 14*sin(3*x) + 2*sin(x) + 1)*sin(5*x)
- 196*sin(5*x)^2 - 120*(7*cos(3*x) - cos(x) - 4*sin(2*x))*sin(4*x) - 900*sin(4*x)^2 - 28*(8*cos(2*x) - 2*sin(x
) - 1)*sin(3*x) - 196*sin(3*x)^2 - 32*cos(x)*sin(2*x) - 64*sin(2*x)^2 - 4*sin(x)^2 - 4*sin(x) - 1), x) + 2*cos
(x))/(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-sin(x)^5),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{\sin ^{5}{\left (x \right )} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-sin(x)**5),x)
[Out]
-Integral(1/(sin(x)**5 - 1), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-sin(x)^5),x, algorithm="giac")
[Out]
sage0*x
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Mupad [B]
time = 14.44, size = 2500, normalized size = 13.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-1/(sin(x)^5 - 1),x)
[Out]
2*atanh((989855744*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(
1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/
5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(
- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)) + (2030043136*tan(x/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)
^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5
- 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*(- (2*5
^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)) + (1627389
952*5^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*t
an(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)
^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5
^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)) + (553648128*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2
)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (
2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 1677
7216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)
) + (184549376*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((3019898
88*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (4
52984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2)
+ (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)) + (5083496448*5^(1/2)*tan(x/2)
*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/12
5 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25
- (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 -
1)^(1/2))/25 - 184549376/25)) + (553648128*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/
50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*
5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 167772
16*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25))
- (553648128*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((
301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/
25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1
)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - 184549376/25)))*((- (2*5^(1/2))/5 - 1)
^(1/2)/50 - 1/50)^(1/2) - 2*atanh((989855744*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((1308622848*
tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (2382364672*5^(1/2)*tan(x/2))/125 - (301989888*tan(x/2))/5 + (45298
4832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (
436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (2030043136*tan(x/2)*(- (- (2*5^
(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (2382364672*5
^(1/2)*tan(x/2))/125 - (301989888*tan(x/2))/5 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216
*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/
25 + 184549376/25)) + (1627389952*5^(1/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((1308622848*ta
n(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (2382364672*5^(1/2)*tan(x/2))/125 - (301989888*tan(x/2))/5 + (4529848
32*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (43
6207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (553648128*(- (2*5^(1/2))/5 - 1)^(
1/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25
- (2382364672*5^(1/2)*tan(x/2))/125 - (301989888*tan(x/2))/5 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)
)/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2)
)/5 - 1)^(1/2))/25 + 184549376/25)) - (184549376*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*(- (- (2*5^(1/2))/5 - 1)^
(1/2)/50 - 1/50)^(1/2))/(5*((1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (2382364672*5^(1/2)*tan(x/2
))/125 - (301989888*tan(x/2))/5 + (452984832*5^...
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