3.73 \(\int \frac {1}{1-\cosh ^5(x)} \, dx\)
Optimal. Leaf size=205 \[ \frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {(-1)^{4/5}-1}}+\frac {2 \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}-\frac {\sinh (x)}{5 (1-\cosh (x))} \]
[Out]
-1/5*sinh(x)/(1-cosh(x))+2/5*arctanh(((1-(-1)^(3/5))/(1+(-1)^(3/5)))^(1/2)*tanh(1/2*x))/(1+(-1)^(1/5))^(1/2)+2
/5*arctanh(((1-(-1)^(1/5))/(1+(-1)^(1/5)))^(1/2)*tanh(1/2*x))/(1-(-1)^(2/5))^(1/2)+2/5*arctan(((-1-(-1)^(4/5))
/(1-(-1)^(4/5)))^(1/2)*tanh(1/2*x))/(-1-(-1)^(3/5))^(1/2)-2/5*arctan(tanh(1/2*x)/((-1+(-1)^(2/5))/(1+(-1)^(2/5
)))^(1/2))/(-1+(-1)^(4/5))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.46, 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, 208, 205} \[ \frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {(-1)^{4/5}-1}}+\frac {2 \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}-\frac {\sinh (x)}{5 (1-\cosh (x))} \]
Antiderivative was successfully verified.
[In]
Int[(1 - Cosh[x]^5)^(-1),x]
[Out]
(-2*ArcTan[Tanh[x/2]/Sqrt[-((1 - (-1)^(2/5))/(1 + (-1)^(2/5)))]])/(5*Sqrt[-1 + (-1)^(4/5)]) + (2*ArcTan[Sqrt[-
((1 + (-1)^(4/5))/(1 - (-1)^(4/5)))]*Tanh[x/2]])/(5*Sqrt[-1 - (-1)^(3/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(1/5))/
(1 + (-1)^(1/5))]*Tanh[x/2]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(3/5))/(1 + (-1)^(3/5))]*Ta
nh[x/2]])/(5*Sqrt[1 + (-1)^(1/5)]) - Sinh[x]/(5*(1 - Cosh[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-\cosh ^5(x)} \, dx &=\int \left (\frac {1}{5 (1-\cosh (x))}+\frac {1}{5 \left (1+\sqrt [5]{-1} \cosh (x)\right )}+\frac {1}{5 \left (1-(-1)^{2/5} \cosh (x)\right )}+\frac {1}{5 \left (1+(-1)^{3/5} \cosh (x)\right )}+\frac {1}{5 \left (1-(-1)^{4/5} \cosh (x)\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{1-\cosh (x)} \, dx+\frac {1}{5} \int \frac {1}{1+\sqrt [5]{-1} \cosh (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{2/5} \cosh (x)} \, dx+\frac {1}{5} \int \frac {1}{1+(-1)^{3/5} \cosh (x)} \, dx+\frac {1}{5} \int \frac {1}{1-(-1)^{4/5} \cosh (x)} \, dx\\ &=-\frac {\sinh (x)}{5 (1-\cosh (x))}+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [5]{-1}-\left (1-\sqrt [5]{-1}\right ) x^2} \, dx,x,\tanh \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,\tanh \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,\tanh \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,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right )}{5 \sqrt {-1+(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {-1-(-1)^{3/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {\sinh (x)}{5 (1-\cosh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.10, size = 445, normalized size = 2.17 \[ \frac {1}{10} \text {RootSum}\left [\text {$\#$1}^8+2 \text {$\#$1}^7+8 \text {$\#$1}^6+14 \text {$\#$1}^5+30 \text {$\#$1}^4+14 \text {$\#$1}^3+8 \text {$\#$1}^2+2 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^6 x+2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+4 \text {$\#$1}^5 x+8 \text {$\#$1}^5 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+15 \text {$\#$1}^4 x+30 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+40 \text {$\#$1}^3 x+80 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+15 \text {$\#$1}^2 x+30 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+4 \text {$\#$1} x+8 \text {$\#$1} \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+x}{4 \text {$\#$1}^7+7 \text {$\#$1}^6+24 \text {$\#$1}^5+35 \text {$\#$1}^4+60 \text {$\#$1}^3+21 \text {$\#$1}^2+8 \text {$\#$1}+1}\& \right ]+\frac {1}{5} \coth \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(1 - Cosh[x]^5)^(-1),x]
[Out]
Coth[x/2]/5 + RootSum[1 + 2*#1 + 8*#1^2 + 14*#1^3 + 30*#1^4 + 14*#1^5 + 8*#1^6 + 2*#1^7 + #1^8 & , (x + 2*Log[
-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + 4*x*#1 + 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 -
Sinh[x/2]*#1]*#1 + 15*x*#1^2 + 30*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 + 40*x*#1^3
+ 80*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3 + 15*x*#1^4 + 30*Log[-Cosh[x/2] - Sinh[x/2
] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + 4*x*#1^5 + 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]
*#1^5 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(1 + 8*#1 + 21*#1^2 + 60*#1
^3 + 35*#1^4 + 24*#1^5 + 7*#1^6 + 4*#1^7) & ]/10
________________________________________________________________________________________
fricas [B] time = 2.12, size = 3260, normalized size = 15.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-cosh(x)^5),x, algorithm="fricas")
[Out]
1/8000*(8*sqrt(10)*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*((sqrt(5) - 1)*e^x - s
qrt(5) + 1)*(40*sqrt(5) + 200)^(1/4)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*arctan(1/40*sqrt(10)*((sqrt(5) - 5)
*e^x - sqrt(5) - 1)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5) + 1/400*sqrt(10)*(sqrt(10)*((sqrt(5) - 5)*e^x - 2*sq
rt(5))*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5) + 10*((sqrt(5) - 1)*e^x + sqrt(5) + 1)*sqrt(2*sqrt(5) + 5))*sqrt(
sqrt(5) + 5) - 1/64000*(80*sqrt(10)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 8*sqrt(10)*(sqrt(10)
*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 10*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 1))*sqrt(sqrt(5) + 5)
- sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*((sqrt(10)*sqrt(2*sqrt(5) + 5)*sqrt(sq
rt(5) + 5)*(sqrt(5) - 3) + 2*(3*sqrt(5) - 7)*sqrt(2*sqrt(5) + 5))*(40*sqrt(5) + 200)^(3/4) + 4*(sqrt(10)*sqrt(
2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 10*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 1))*(40*sqrt(5) + 200)^(1/4
)) + 1600*sqrt(2*sqrt(5) + 5))*sqrt(-20*sqrt(10)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 200*(sqrt(5) + 1)*e^x + 2*(
sqrt(10)*(sqrt(5)*e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 5*sqrt(5) + 25)*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(
sqrt(5) + 5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4) + 400*e^(2*x) + 400) - 1/16000*sqrt(2*sqrt(10)*(2*sqr
t(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*((sqrt(10)*(5*(sqrt(5) - 3)*e^x - 2*sqrt(5))*sqrt(2*sqrt(5) + 5
)*sqrt(sqrt(5) + 5) + 10*((3*sqrt(5) - 7)*e^x + 2)*sqrt(2*sqrt(5) + 5))*(40*sqrt(5) + 200)^(3/4) + 20*(sqrt(10
)*((sqrt(5) - 5)*e^x - 2*sqrt(5))*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5) + 10*((sqrt(5) - 1)*e^x + sqrt(5) + 1)
*sqrt(2*sqrt(5) + 5))*(40*sqrt(5) + 200)^(1/4)) + 1/4*sqrt(2*sqrt(5) + 5)*(sqrt(5) + 2*e^x + 1)) + 8*sqrt(10)*
sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*((sqrt(5) - 1)*e^x - sqrt(5) + 1)*(40*sqr
t(5) + 200)^(1/4)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*arctan(-1/40*sqrt(10)*((sqrt(5) - 5)*e^x - sqrt(5) - 1
)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 1/400*sqrt(10)*(sqrt(10)*((sqrt(5) - 5)*e^x - 2*sqrt(5))*sqrt(2*sqrt
(5) + 5)*sqrt(sqrt(5) + 5) + 10*((sqrt(5) - 1)*e^x + sqrt(5) + 1)*sqrt(2*sqrt(5) + 5))*sqrt(sqrt(5) + 5) + 1/6
4000*(80*sqrt(10)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 8*sqrt(10)*(sqrt(10)*sqrt(2*sqrt(5) +
5)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 10*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 1))*sqrt(sqrt(5) + 5) + sqrt(2*sqrt(10)
*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*((sqrt(10)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5)*(sqrt(5
) - 3) + 2*(3*sqrt(5) - 7)*sqrt(2*sqrt(5) + 5))*(40*sqrt(5) + 200)^(3/4) + 4*(sqrt(10)*sqrt(2*sqrt(5) + 5)*sqr
t(sqrt(5) + 5)*(sqrt(5) - 5) + 10*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 1))*(40*sqrt(5) + 200)^(1/4)) + 1600*sqrt(2*s
qrt(5) + 5))*sqrt(-20*sqrt(10)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 200*(sqrt(5) + 1)*e^x - 2*(sqrt(10)*(sqrt(5)*
e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 5*sqrt(5) + 25)*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*
sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4) + 400*e^(2*x) + 400) - 1/16000*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqr
t(5) + 5) - 20*sqrt(5) + 60)*((sqrt(10)*(5*(sqrt(5) - 3)*e^x - 2*sqrt(5))*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5
) + 10*((3*sqrt(5) - 7)*e^x + 2)*sqrt(2*sqrt(5) + 5))*(40*sqrt(5) + 200)^(3/4) + 20*(sqrt(10)*((sqrt(5) - 5)*e
^x - 2*sqrt(5))*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 5) + 10*((sqrt(5) - 1)*e^x + sqrt(5) + 1)*sqrt(2*sqrt(5) +
5))*(40*sqrt(5) + 200)^(1/4)) - 1/4*sqrt(2*sqrt(5) + 5)*(sqrt(5) + 2*e^x + 1)) + 4*((sqrt(5) + 1)*e^x - sqrt(5
) - 1)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*sqrt(-2*sqrt(5) + 5)*(-40*sqrt(5) + 20
0)^(3/4)*arctan(-1/32000*((20*(3*sqrt(5) + 7)*e^x + (5*(sqrt(5) + 3)*e^x - 2*sqrt(5))*sqrt(-40*sqrt(5) + 200)
- 40)*(-40*sqrt(5) + 200)^(3/4) + 20*(20*(sqrt(5) + 1)*e^x + ((sqrt(5) + 5)*e^x - 2*sqrt(5))*sqrt(-40*sqrt(5)
+ 200) + 20*sqrt(5) - 20)*(-40*sqrt(5) + 200)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5
) + 60)*sqrt(-2*sqrt(5) + 5) + 1/128000*((((sqrt(5) + 3)*sqrt(-40*sqrt(5) + 200) + 12*sqrt(5) + 28)*(-40*sqrt(
5) + 200)^(3/4) + 4*((sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 20)*(-40*sqrt(5) + 200)^(1/4))*sqrt(
-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*sqrt(-2*sqrt(5) + 5) + 4*(((sqrt(5) + 5)*sqrt(-40*
sqrt(5) + 200) + 20*sqrt(5) + 20)*sqrt(-40*sqrt(5) + 200) + 20*(sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) - 800)*sq
rt(-2*sqrt(5) + 5))*sqrt(-200*(sqrt(5) - 1)*e^x + ((sqrt(5)*e^x - sqrt(5) - 5)*sqrt(-40*sqrt(5) + 200) - 10*sq
rt(5) - 50)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*(-40*sqrt(5) + 200)^(1/4) + 10*(s
qrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 400*e^(2*x) + 400) - 1/1600*(20*((sqrt(5) + 5)*e^x - sqrt(5) + 1)*sqrt(-
40*sqrt(5) + 200) + (20*(sqrt(5) + 1)*e^x + ((sqrt(5) + 5)*e^x - 2*sqrt(5))*sqrt(-40*sqrt(5) + 200) + 20*sqrt(
5) - 20)*sqrt(-40*sqrt(5) + 200) + 400*sqrt(5) - 800*e^x - 400)*sqrt(-2*sqrt(5) + 5)) + 4*((sqrt(5) + 1)*e^x -
sqrt(5) - 1)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*sqrt(-2*sqrt(5) + 5)*(-40*sqrt(
5) + 200)^(3/4)*arctan(-1/32000*((20*(3*sqrt(5) + 7)*e^x + (5*(sqrt(5) + 3)*e^x - 2*sqrt(5))*sqrt(-40*sqrt(5)
+ 200) - 40)*(-40*sqrt(5) + 200)^(3/4) + 20*(20*(sqrt(5) + 1)*e^x + ((sqrt(5) + 5)*e^x - 2*sqrt(5))*sqrt(-40*s
qrt(5) + 200) + 20*sqrt(5) - 20)*(-40*sqrt(5) + 200)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20
*sqrt(5) + 60)*sqrt(-2*sqrt(5) + 5) + 1/128000*((((sqrt(5) + 3)*sqrt(-40*sqrt(5) + 200) + 12*sqrt(5) + 28)*(-4
0*sqrt(5) + 200)^(3/4) + 4*((sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 20)*(-40*sqrt(5) + 200)^(1/4)
)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*sqrt(-2*sqrt(5) + 5) - 4*(((sqrt(5) + 5)*sq
rt(-40*sqrt(5) + 200) + 20*sqrt(5) + 20)*sqrt(-40*sqrt(5) + 200) + 20*(sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) -
800)*sqrt(-2*sqrt(5) + 5))*sqrt(-200*(sqrt(5) - 1)*e^x - ((sqrt(5)*e^x - sqrt(5) - 5)*sqrt(-40*sqrt(5) + 200)
- 10*sqrt(5) - 50)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*(-40*sqrt(5) + 200)^(1/4)
+ 10*(sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 400*e^(2*x) + 400) + 1/1600*(20*((sqrt(5) + 5)*e^x - sqrt(5) + 1)
*sqrt(-40*sqrt(5) + 200) + (20*(sqrt(5) + 1)*e^x + ((sqrt(5) + 5)*e^x - 2*sqrt(5))*sqrt(-40*sqrt(5) + 200) + 2
0*sqrt(5) - 20)*sqrt(-40*sqrt(5) + 200) + 400*sqrt(5) - 800*e^x - 400)*sqrt(-2*sqrt(5) + 5)) + 2*(sqrt(10)*((s
qrt(5) - 5)*e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 40*e^x + 40)*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5) +
5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4)*log(-20*sqrt(10)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 200*(sqrt(5
) + 1)*e^x + 2*(sqrt(10)*(sqrt(5)*e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 5*sqrt(5) + 25)*sqrt(2*sqrt(10)*(2*sq
rt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4) + 400*e^(2*x) + 400) - 2*(sqrt(10)*((
sqrt(5) - 5)*e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 40*e^x + 40)*sqrt(2*sqrt(10)*(2*sqrt(5) - 5)*sqrt(sqrt(5)
+ 5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4)*log(-20*sqrt(10)*sqrt(sqrt(5) + 5)*(sqrt(5) - 5) + 200*(sqrt(
5) + 1)*e^x - 2*(sqrt(10)*(sqrt(5)*e^x - sqrt(5) + 5)*sqrt(sqrt(5) + 5) - 5*sqrt(5) + 25)*sqrt(2*sqrt(10)*(2*s
qrt(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4) + 400*e^(2*x) + 400) + (((sqrt(5) +
5)*e^x - sqrt(5) - 5)*sqrt(-40*sqrt(5) + 200) + 80*e^x - 80)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 2
0*sqrt(5) + 60)*(-40*sqrt(5) + 200)^(1/4)*log(-200*(sqrt(5) - 1)*e^x + ((sqrt(5)*e^x - sqrt(5) - 5)*sqrt(-40*s
qrt(5) + 200) - 10*sqrt(5) - 50)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*(-40*sqrt(5)
+ 200)^(1/4) + 10*(sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 400*e^(2*x) + 400) - (((sqrt(5) + 5)*e^x - sqrt(5)
- 5)*sqrt(-40*sqrt(5) + 200) + 80*e^x - 80)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*(
-40*sqrt(5) + 200)^(1/4)*log(-200*(sqrt(5) - 1)*e^x - ((sqrt(5)*e^x - sqrt(5) - 5)*sqrt(-40*sqrt(5) + 200) - 1
0*sqrt(5) - 50)*sqrt(-(2*sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 20*sqrt(5) + 60)*(-40*sqrt(5) + 200)^(1/4) + 1
0*(sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 400*e^(2*x) + 400) + 3200)/(e^x - 1)
________________________________________________________________________________________
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-cosh(x)^5),x, algorithm="giac")
[Out]
sage0*x
________________________________________________________________________________________
maple [C] time = 0.07, size = 64, normalized size = 0.31 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+10 \textit {\_Z}^{4}+5\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+5 \textit {\_R}^{4}-5 \textit {\_R}^{2}+5\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+5 \textit {\_R}^{3}}\right )}{10}+\frac {1}{5 \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1-cosh(x)^5),x)
[Out]
1/10*sum((-_R^6+5*_R^4-5*_R^2+5)/(_R^7+5*_R^3)*ln(tanh(1/2*x)-_R),_R=RootOf(_Z^8+10*_Z^4+5))+1/5/tanh(1/2*x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{5 \, {\left (e^{x} - 1\right )}} + \int \frac {2 \, {\left (e^{\left (7 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 15 \, e^{\left (5 \, x\right )} + 40 \, e^{\left (4 \, x\right )} + 15 \, e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{5 \, {\left (e^{\left (8 \, x\right )} + 2 \, e^{\left (7 \, x\right )} + 8 \, e^{\left (6 \, x\right )} + 14 \, e^{\left (5 \, x\right )} + 30 \, e^{\left (4 \, x\right )} + 14 \, e^{\left (3 \, x\right )} + 8 \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-cosh(x)^5),x, algorithm="maxima")
[Out]
2/5/(e^x - 1) + integrate(2/5*(e^(7*x) + 4*e^(6*x) + 15*e^(5*x) + 40*e^(4*x) + 15*e^(3*x) + 4*e^(2*x) + e^x)/(
e^(8*x) + 2*e^(7*x) + 8*e^(6*x) + 14*e^(5*x) + 30*e^(4*x) + 14*e^(3*x) + 8*e^(2*x) + 2*e^x + 1), x)
________________________________________________________________________________________
mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-1/(cosh(x)^5 - 1),x)
[Out]
\text{Hanged}
________________________________________________________________________________________
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-cosh(x)**5),x)
[Out]
Timed out
________________________________________________________________________________________