3.72 \(\int \frac{1}{1+\cosh ^8(x)} \, dx\)
Optimal. Leaf size=129 \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-\sqrt [4]{-1}}}\right )}{4 \sqrt{1-\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+\sqrt [4]{-1}}}\right )}{4 \sqrt{1+\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-(-1)^{3/4}}}\right )}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+(-1)^{3/4}}}\right )}{4 \sqrt{1+(-1)^{3/4}}} \]
[Out]
ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/4)]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(1/4)]]/(4*Sqrt
[1 + (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(3/4)]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTanh[Tanh[x]/Sqrt[1 +
(-1)^(3/4)]]/(4*Sqrt[1 + (-1)^(3/4)])
________________________________________________________________________________________
Rubi [A] time = 0.18512, antiderivative size = 129, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used =
{3211, 3181, 206} \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-\sqrt [4]{-1}}}\right )}{4 \sqrt{1-\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+\sqrt [4]{-1}}}\right )}{4 \sqrt{1+\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-(-1)^{3/4}}}\right )}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+(-1)^{3/4}}}\right )}{4 \sqrt{1+(-1)^{3/4}}} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Cosh[x]^8)^(-1),x]
[Out]
ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/4)]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(1/4)]]/(4*Sqrt
[1 + (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(3/4)]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTanh[Tanh[x]/Sqrt[1 +
(-1)^(3/4)]]/(4*Sqrt[1 + (-1)^(3/4)])
Rule 3211
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]
Rule 3181
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{1}{1+\cosh ^8(x)} \, dx &=\frac{1}{4} \int \frac{1}{1-\sqrt [4]{-1} \cosh ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1+\sqrt [4]{-1} \cosh ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1-(-1)^{3/4} \cosh ^2(x)} \, dx+\frac{1}{4} \int \frac{1}{1+(-1)^{3/4} \cosh ^2(x)} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\left (1-\sqrt [4]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\left (1+\sqrt [4]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\left (1-(-1)^{3/4}\right ) x^2} \, dx,x,\coth (x)\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\left (1+(-1)^{3/4}\right ) x^2} \, dx,x,\coth (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-\sqrt [4]{-1}}}\right )}{4 \sqrt{1-\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+\sqrt [4]{-1}}}\right )}{4 \sqrt{1+\sqrt [4]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-(-1)^{3/4}}}\right )}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+(-1)^{3/4}}}\right )}{4 \sqrt{1+(-1)^{3/4}}}\\ \end{align*}
Mathematica [C] time = 0.128979, size = 127, normalized size = 0.98 \[ 16 \text{RootSum}\left [\text{$\#$1}^8+8 \text{$\#$1}^7+28 \text{$\#$1}^6+56 \text{$\#$1}^5+326 \text{$\#$1}^4+56 \text{$\#$1}^3+28 \text{$\#$1}^2+8 \text{$\#$1}+1\& ,\frac{\text{$\#$1}^3 x+\text{$\#$1}^3 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text{$\#$1}^7+7 \text{$\#$1}^6+21 \text{$\#$1}^5+35 \text{$\#$1}^4+163 \text{$\#$1}^3+21 \text{$\#$1}^2+7 \text{$\#$1}+1}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + Cosh[x]^8)^(-1),x]
[Out]
16*RootSum[1 + 8*#1 + 28*#1^2 + 56*#1^3 + 326*#1^4 + 56*#1^5 + 28*#1^6 + 8*#1^7 + #1^8 & , (x*#1^3 + Log[-Cosh
[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^3)/(1 + 7*#1 + 21*#1^2 + 163*#1^3 + 35*#1^4 + 21*#1^5 + 7*#1^6 + #
1^7) & ]
________________________________________________________________________________________
Maple [C] time = 0.029, size = 47, normalized size = 0.4 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{8}-4\,{{\it \_Z}}^{6}+6\,{{\it \_Z}}^{4}-4\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ( 2\,{\it \_R}\,\tanh \left ( x/2 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+cosh(x)^8),x)
[Out]
1/8*sum(_R*ln(2*_R*tanh(1/2*x)+tanh(1/2*x)^2+1),_R=RootOf(2*_Z^8-4*_Z^6+6*_Z^4-4*_Z^2+1))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cosh \left (x\right )^{8} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+cosh(x)^8),x, algorithm="maxima")
[Out]
integrate(1/(cosh(x)^8 + 1), x)
________________________________________________________________________________________
Fricas [B] time = 4.02447, size = 11980, normalized size = 92.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+cosh(x)^8),x, algorithm="fricas")
[Out]
-1/16*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4)*sqrt(2*sqrt(2) + 3)*(s
qrt(2) - 1)*arctan(1/31*(2*(13*sqrt(2) - 20)*e^(2*x) - 23*sqrt(2) + 33)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3
) + 1/496*(32*(10*sqrt(2) - 13)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + (((355*sqrt(2) - 508)*sqrt(2*sqrt(2)
+ 4)*sqrt(2*sqrt(2) + 3) + 6*(59*sqrt(2) - 86)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) + 4)^(3/4) + 4*((82*sqrt(2) -
119)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + (85*sqrt(2) - 126)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) + 4)^(1/4))*
sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8) + 4*((76*sqrt(2) - 105)*sqrt(2*sqrt(2) + 4)*sqrt(2
*sqrt(2) + 3) + 2*(53*sqrt(2) - 72)*sqrt(2*sqrt(2) + 3))*sqrt(2*sqrt(2) + 4) + 16*(23*sqrt(2) - 33)*sqrt(2*sqr
t(2) + 3))*sqrt(-4*(sqrt(2) - 1)*e^(2*x) + (2*(sqrt(2) - 1)*e^(2*x) + ((sqrt(2) - 2)*e^(2*x) + 5*sqrt(2) - 6)*
sqrt(2*sqrt(2) + 4) + 6*sqrt(2) - 6)*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(2*sqrt(2) +
4)^(1/4) - 4*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) - 4*sqrt(2) + 2*e^(4*x) + 10) + 1/248*((((254*sqrt(2) - 355)*e^
(2*x) + 102*sqrt(2) - 145)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 2*(3*(43*sqrt(2) - 59)*e^(2*x) + 23*sqrt(
2) - 33)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) + 4)^(3/4) + 2*(((119*sqrt(2) - 164)*e^(2*x) + 39*sqrt(2) - 60)*sqrt(
2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 2*((63*sqrt(2) - 85)*e^(2*x) + 17*sqrt(2) - 19)*sqrt(2*sqrt(2) + 3))*(2*s
qrt(2) + 4)^(1/4))*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8) + 1/124*(((105*sqrt(2) - 152)*e
^(2*x) - 13*sqrt(2) + 20)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 4*((36*sqrt(2) - 53)*e^(2*x) + 23*sqrt(2)
- 33)*sqrt(2*sqrt(2) + 3))*sqrt(2*sqrt(2) + 4) + 1/31*((33*sqrt(2) - 46)*e^(2*x) + 3*sqrt(2) - 7)*sqrt(2*sqrt(
2) + 3)) - 1/16*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4)*sqrt(2*sqrt(
2) + 3)*(sqrt(2) - 1)*arctan(-1/31*(2*(13*sqrt(2) - 20)*e^(2*x) - 23*sqrt(2) + 33)*sqrt(2*sqrt(2) + 4)*sqrt(2*
sqrt(2) + 3) - 1/496*(32*(10*sqrt(2) - 13)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) - (((355*sqrt(2) - 508)*sqr
t(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 6*(59*sqrt(2) - 86)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) + 4)^(3/4) + 4*((82
*sqrt(2) - 119)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + (85*sqrt(2) - 126)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) +
4)^(1/4))*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8) + 4*((76*sqrt(2) - 105)*sqrt(2*sqrt(2)
+ 4)*sqrt(2*sqrt(2) + 3) + 2*(53*sqrt(2) - 72)*sqrt(2*sqrt(2) + 3))*sqrt(2*sqrt(2) + 4) + 16*(23*sqrt(2) - 33)
*sqrt(2*sqrt(2) + 3))*sqrt(-4*(sqrt(2) - 1)*e^(2*x) - (2*(sqrt(2) - 1)*e^(2*x) + ((sqrt(2) - 2)*e^(2*x) + 5*sq
rt(2) - 6)*sqrt(2*sqrt(2) + 4) + 6*sqrt(2) - 6)*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(2
*sqrt(2) + 4)^(1/4) - 4*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) - 4*sqrt(2) + 2*e^(4*x) + 10) + 1/248*((((254*sqrt(2
) - 355)*e^(2*x) + 102*sqrt(2) - 145)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 2*(3*(43*sqrt(2) - 59)*e^(2*x)
+ 23*sqrt(2) - 33)*sqrt(2*sqrt(2) + 3))*(2*sqrt(2) + 4)^(3/4) + 2*(((119*sqrt(2) - 164)*e^(2*x) + 39*sqrt(2)
- 60)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 2*((63*sqrt(2) - 85)*e^(2*x) + 17*sqrt(2) - 19)*sqrt(2*sqrt(2)
+ 3))*(2*sqrt(2) + 4)^(1/4))*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8) - 1/124*(((105*sqrt(
2) - 152)*e^(2*x) - 13*sqrt(2) + 20)*sqrt(2*sqrt(2) + 4)*sqrt(2*sqrt(2) + 3) + 4*((36*sqrt(2) - 53)*e^(2*x) +
23*sqrt(2) - 33)*sqrt(2*sqrt(2) + 3))*sqrt(2*sqrt(2) + 4) - 1/31*((33*sqrt(2) - 46)*e^(2*x) + 3*sqrt(2) - 7)*s
qrt(2*sqrt(2) + 3)) + 1/16*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*(sqrt(2) + 1)*(-2*sqr
t(2) + 4)^(3/4)*sqrt(-2*sqrt(2) + 3)*arctan(1/31*(2*(13*sqrt(2) + 20)*e^(2*x) - 23*sqrt(2) - 33)*sqrt(-2*sqrt(
2) + 4)*sqrt(-2*sqrt(2) + 3) - 1/496*(32*(10*sqrt(2) + 13)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) - (((355*
sqrt(2) + 508)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 6*(59*sqrt(2) + 86)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(
2) + 4)^(3/4) + 4*((82*sqrt(2) + 119)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + (85*sqrt(2) + 126)*sqrt(-2*s
qrt(2) + 3))*(-2*sqrt(2) + 4)^(1/4))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*((76*sq
rt(2) + 105)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 2*(53*sqrt(2) + 72)*sqrt(-2*sqrt(2) + 3))*sqrt(-2*sqr
t(2) + 4) + 16*(23*sqrt(2) + 33)*sqrt(-2*sqrt(2) + 3))*sqrt(4*(sqrt(2) + 1)*e^(2*x) + (2*(sqrt(2) + 1)*e^(2*x)
+ ((sqrt(2) + 2)*e^(2*x) + 5*sqrt(2) + 6)*sqrt(-2*sqrt(2) + 4) + 6*sqrt(2) + 6)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(
-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4) + 4*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 2
*e^(4*x) + 10) - 1/248*((((254*sqrt(2) + 355)*e^(2*x) + 102*sqrt(2) + 145)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2
) + 3) + 2*(3*(43*sqrt(2) + 59)*e^(2*x) + 23*sqrt(2) + 33)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(2) + 4)^(3/4) + 2*((
(119*sqrt(2) + 164)*e^(2*x) + 39*sqrt(2) + 60)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 2*((63*sqrt(2) + 85
)*e^(2*x) + 17*sqrt(2) + 19)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(2) + 4)^(1/4))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqr
t(2) + 4) + 4*sqrt(2) + 8) + 1/124*(((105*sqrt(2) + 152)*e^(2*x) - 13*sqrt(2) - 20)*sqrt(-2*sqrt(2) + 4)*sqrt(
-2*sqrt(2) + 3) + 4*((36*sqrt(2) + 53)*e^(2*x) + 23*sqrt(2) + 33)*sqrt(-2*sqrt(2) + 3))*sqrt(-2*sqrt(2) + 4) +
1/31*((33*sqrt(2) + 46)*e^(2*x) + 3*sqrt(2) + 7)*sqrt(-2*sqrt(2) + 3)) + 1/16*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2
*sqrt(2) + 4) + 4*sqrt(2) + 8)*(sqrt(2) + 1)*(-2*sqrt(2) + 4)^(3/4)*sqrt(-2*sqrt(2) + 3)*arctan(-1/31*(2*(13*s
qrt(2) + 20)*e^(2*x) - 23*sqrt(2) - 33)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 1/496*(32*(10*sqrt(2) + 13
)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + (((355*sqrt(2) + 508)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3)
+ 6*(59*sqrt(2) + 86)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(2) + 4)^(3/4) + 4*((82*sqrt(2) + 119)*sqrt(-2*sqrt(2) + 4
)*sqrt(-2*sqrt(2) + 3) + (85*sqrt(2) + 126)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(2) + 4)^(1/4))*sqrt(-2*(2*sqrt(2) +
3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*((76*sqrt(2) + 105)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) +
2*(53*sqrt(2) + 72)*sqrt(-2*sqrt(2) + 3))*sqrt(-2*sqrt(2) + 4) + 16*(23*sqrt(2) + 33)*sqrt(-2*sqrt(2) + 3))*sq
rt(4*(sqrt(2) + 1)*e^(2*x) - (2*(sqrt(2) + 1)*e^(2*x) + ((sqrt(2) + 2)*e^(2*x) + 5*sqrt(2) + 6)*sqrt(-2*sqrt(2
) + 4) + 6*sqrt(2) + 6)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4) +
4*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 2*e^(4*x) + 10) - 1/248*((((254*sqrt(2) + 355)*e^(2*x) + 1
02*sqrt(2) + 145)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 2*(3*(43*sqrt(2) + 59)*e^(2*x) + 23*sqrt(2) + 33
)*sqrt(-2*sqrt(2) + 3))*(-2*sqrt(2) + 4)^(3/4) + 2*(((119*sqrt(2) + 164)*e^(2*x) + 39*sqrt(2) + 60)*sqrt(-2*sq
rt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 2*((63*sqrt(2) + 85)*e^(2*x) + 17*sqrt(2) + 19)*sqrt(-2*sqrt(2) + 3))*(-2*sq
rt(2) + 4)^(1/4))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) - 1/124*(((105*sqrt(2) + 152)*
e^(2*x) - 13*sqrt(2) - 20)*sqrt(-2*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 3) + 4*((36*sqrt(2) + 53)*e^(2*x) + 23*sqrt(
2) + 33)*sqrt(-2*sqrt(2) + 3))*sqrt(-2*sqrt(2) + 4) - 1/31*((33*sqrt(2) + 46)*e^(2*x) + 3*sqrt(2) + 7)*sqrt(-2
*sqrt(2) + 3)) - 1/64*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt(2) + 1)*sqrt(-2*sqr
t(2) + 4) + 2*sqrt(2))*(-2*sqrt(2) + 4)^(1/4)*log(2*(sqrt(2) + 1)*e^(2*x) + 1/2*(2*(sqrt(2) + 1)*e^(2*x) + ((s
qrt(2) + 2)*e^(2*x) + 5*sqrt(2) + 6)*sqrt(-2*sqrt(2) + 4) + 6*sqrt(2) + 6)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqr
t(2) + 4) + 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4) + 2*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + e^(4*x)
+ 5) + 1/64*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*((sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4)
+ 2*sqrt(2))*(-2*sqrt(2) + 4)^(1/4)*log(2*(sqrt(2) + 1)*e^(2*x) - 1/2*(2*(sqrt(2) + 1)*e^(2*x) + ((sqrt(2) +
2)*e^(2*x) + 5*sqrt(2) + 6)*sqrt(-2*sqrt(2) + 4) + 6*sqrt(2) + 6)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4)
+ 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4) + 2*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + e^(4*x) + 5) - 1
/64*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1) + 2*sqrt(2)
)*(2*sqrt(2) + 4)^(1/4)*log(-2*(sqrt(2) - 1)*e^(2*x) + 1/2*(2*(sqrt(2) - 1)*e^(2*x) + ((sqrt(2) - 2)*e^(2*x) +
5*sqrt(2) - 6)*sqrt(2*sqrt(2) + 4) + 6*sqrt(2) - 6)*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) +
8)*(2*sqrt(2) + 4)^(1/4) - 2*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) - 2*sqrt(2) + e^(4*x) + 5) + 1/64*sqrt(2*sqrt(2
*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1) + 2*sqrt(2))*(2*sqrt(2) + 4)
^(1/4)*log(-2*(sqrt(2) - 1)*e^(2*x) - 1/2*(2*(sqrt(2) - 1)*e^(2*x) + ((sqrt(2) - 2)*e^(2*x) + 5*sqrt(2) - 6)*s
qrt(2*sqrt(2) + 4) + 6*sqrt(2) - 6)*sqrt(2*sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) - 4*sqrt(2) + 8)*(2*sqrt(2) + 4
)^(1/4) - 2*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) - 2*sqrt(2) + e^(4*x) + 5)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+cosh(x)**8),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.37489, size = 1, normalized size = 0.01 \begin{align*} 0 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+cosh(x)^8),x, algorithm="giac")
[Out]
0