3.70 \(\int \frac {1}{1+\cosh ^5(x)} \, dx\)
Optimal. Leaf size=223 \[ \frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {(-1)^{2/5}-1}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {\sinh (x)}{5 (\cosh (x)+1)} \]
[Out]
1/5*sinh(x)/(1+cosh(x))-2/5*arctan(tanh(1/2*x)/((-1+(-1)^(1/5))/(1+(-1)^(1/5)))^(1/2))/(-1+(-1)^(2/5))^(1/2)+2
/5*arctanh(((1-(-1)^(4/5))/(1+(-1)^(4/5)))^(1/2)*tanh(1/2*x))/(1+(-1)^(3/5))^(1/2)-2/5*arctan(((-1-(-1)^(3/5))
/(1-(-1)^(3/5)))^(1/2)*tanh(1/2*x))*((-1-(-1)^(3/5))/(1-(-1)^(3/5)))^(1/2)/(1+(-1)^(3/5))+2/5*arctanh(((1-(-1)
^(2/5))/(1+(-1)^(2/5)))^(1/2)*tanh(1/2*x))/(1-(-1)^(4/5))^(1/2)
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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, 205, 208} \[ \frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {(-1)^{2/5}-1}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {\sinh (x)}{5 (\cosh (x)+1)} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Cosh[x]^5)^(-1),x]
[Out]
(-2*ArcTan[Tanh[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)))]*ArcTan[Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*Tanh[x/2]])/(5*(1 + (-1)^(3/5)
)) + (2*ArcTanh[Sqrt[(1 - (-1)^(2/5))/(1 + (-1)^(2/5))]*Tanh[x/2]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTanh[Sqrt
[(1 - (-1)^(4/5))/(1 + (-1)^(4/5))]*Tanh[x/2]])/(5*Sqrt[1 + (-1)^(3/5)]) + 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\\ &=-\left (\frac {1}{5} \int \frac {1}{-1-\cosh (x)} \, dx\right )-\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-\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}}} \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-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))}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 445, normalized size = 2.00 \[ \frac {1}{5} \tanh \left (\frac {x}{2}\right )-\frac {1}{10} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^7+8 \text {$\#$1}^6-14 \text {$\#$1}^5+30 \text {$\#$1}^4-14 \text {$\#$1}^3+8 \text {$\#$1}^2-2 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^6 x+2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-4 \text {$\#$1}^5 x-8 \text {$\#$1}^5 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+15 \text {$\#$1}^4 x+30 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-40 \text {$\#$1}^3 x-80 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+15 \text {$\#$1}^2 x+30 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-4 \text {$\#$1} x-8 \text {$\#$1} \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+x}{4 \text {$\#$1}^7-7 \text {$\#$1}^6+24 \text {$\#$1}^5-35 \text {$\#$1}^4+60 \text {$\#$1}^3-21 \text {$\#$1}^2+8 \text {$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + Cosh[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 & , (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) & ] + Tanh[x/2]/5
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fricas [B] time = 1.05, size = 3228, normalized size = 14.48 \[ \text {result too large to display} \]
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 +
sqrt(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*s
qrt(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(s
qrt(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*sq
rt(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(1
0)*((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*sq
rt(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*sqr
t(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(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)*sq
rt(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*sqrt(5) - 5)*sqrt(sq
rt(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) + 2
00)^(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)*s
qrt(-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) - 20
*sqrt(5) + 20)*sqrt(-40*sqrt(5) + 200) - 400*sqrt(5) - 800*e^x + 400)*sqrt(-2*sqrt(5) + 5)) + 2*(sqrt(10)*((sq
rt(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*sqr
t(5) - 5)*sqrt(sqrt(5) + 5) - 20*sqrt(5) + 60)*(40*sqrt(5) + 200)^(1/4) + 400*e^(2*x) + 400) - 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) + (((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*sqr
t(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)*(-4
0*sqrt(5) + 200)^(1/4)*log(200*(sqrt(5) - 1)*e^x - ((sqrt(5)*e^x + sqrt(5) + 5)*sqrt(-40*sqrt(5) + 200) + 10*s
qrt(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) + 3200)/(e^x + 1)
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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+cosh(x)^5),x, algorithm="giac")
[Out]
sage0*x
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maple [C] time = 0.07, size = 62, normalized size = 0.28 \[ \frac {\tanh \left (\frac {x}{2}\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (5 \textit {\_Z}^{8}+10 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-5 \textit {\_R}^{6}+5 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+\textit {\_R}^{3}}\right )}{50} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+cosh(x)^5),x)
[Out]
1/5*tanh(1/2*x)+1/50*sum((-5*_R^6+5*_R^4-5*_R^2+1)/(_R^7+_R^3)*ln(tanh(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 \[ -\frac {2}{5 \, {\left (e^{x} + 1\right )}} - \int \frac {2 \, {\left (e^{\left (7 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 15 \, e^{\left (5 \, x\right )} - 40 \, e^{\left (4 \, x\right )} + 15 \, e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{5 \, {\left (e^{\left (8 \, x\right )} - 2 \, e^{\left (7 \, x\right )} + 8 \, e^{\left (6 \, x\right )} - 14 \, e^{\left (5 \, x\right )} + 30 \, e^{\left (4 \, x\right )} - 14 \, e^{\left (3 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}\,{d x} \]
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)
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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}
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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+cosh(x)**5),x)
[Out]
Timed out
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