3.1.72 \(\int \frac {1}{1+\cosh ^8(x)} \, dx\) [72]
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]
1/4*arctanh(tanh(x)/(1-(-1)^(1/4))^(1/2))/(1-(-1)^(1/4))^(1/2)+1/4*arctanh(tanh(x)/(1+(-1)^(1/4))^(1/2))/(1+(-
1)^(1/4))^(1/2)+1/4*arctanh(tanh(x)/(1-(-1)^(3/4))^(1/2))/(1-(-1)^(3/4))^(1/2)+1/4*arctanh(tanh(x)/(1+(-1)^(3/
4))^(1/2))/(1+(-1)^(3/4))^(1/2)
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Rubi [A]
time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, 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 = {3290, 3260, 212}
\begin {gather*} \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 {gather*}
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 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 3260
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 3290
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]
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} \text {Subst}\left (\int \frac {1}{1-\left (1-\sqrt [4]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\left (1+\sqrt [4]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\left (1-(-1)^{3/4}\right ) x^2} \, dx,x,\coth (x)\right )+\frac {1}{4} \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.09, size = 127, normalized size = 0.98 \begin {gather*} 16 \text {RootSum}\left [1+8 \text {$\#$1}+28 \text {$\#$1}^2+56 \text {$\#$1}^3+326 \text {$\#$1}^4+56 \text {$\#$1}^5+28 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {x \text {$\#$1}^3+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^3}{1+7 \text {$\#$1}+21 \text {$\#$1}^2+163 \text {$\#$1}^3+35 \text {$\#$1}^4+21 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \end {gather*}
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) & ]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.59, size = 47, normalized size = 0.36
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2 \tanh \left (\frac {x}{2}\right ) \textit {\_R} +\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\right )}{8}\) |
\(47\) |
| risch |
\(\munderset {\textit {\_R} =\RootOf \left (33554432 \textit {\_Z}^{8}-1048576 \textit {\_Z}^{6}+24576 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-8388608 \textit {\_R}^{7}+1048576 \textit {\_R}^{6}+131072 \textit {\_R}^{5}-16384 \textit {\_R}^{4}-4096 \textit {\_R}^{3}+512 \textit {\_R}^{2}+{\mathrm e}^{2 x}-1\right )\) |
\(66\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+cosh(x)^8),x,method=_RETURNVERBOSE)
[Out]
1/8*sum(_R*ln(2*tanh(1/2*x)*_R+tanh(1/2*x)^2+1),_R=RootOf(2*_Z^8-4*_Z^6+6*_Z^4-4*_Z^2+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+cosh(x)^8),x, algorithm="maxima")
[Out]
integrate(1/(cosh(x)^8 + 1), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3773 vs.
\(2 (89) = 178\).
time = 0.51, size = 3773, normalized size = 29.25 \begin {gather*} \text {Too large to display} \end {gather*}
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) + ...
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Sympy [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+cosh(x)**8),x)
[Out]
Timed out
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Giac [A]
time = 0.43, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+cosh(x)^8),x, algorithm="giac")
[Out]
0
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cosh(x)^8 + 1),x)
[Out]
\text{Hanged}
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