3.1.74 \(\int \frac {1}{1-\cosh ^6(x)} \, dx\) [74]

Optimal. Leaf size=71 \[ \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\coth (x)}{3} \]

[Out]

1/3*coth(x)+1/3*arctanh(tanh(x)/(1+(-1)^(1/3))^(1/2))/(1+(-1)^(1/3))^(1/2)+1/3*arctanh(tanh(x)/(1-(-1)^(2/3))^
(1/2))/(1-(-1)^(2/3))^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3290, 3260, 212, 3254, 3852, 8} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\coth (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Cosh[x]^6)^(-1),x]

[Out]

ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(1/3)]]/(3*Sqrt[1 + (-1)^(1/3)]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(2/3)]]/(3*Sqrt
[1 - (-1)^(2/3)]) + Coth[x]/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 3254

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {1}{1-\cosh ^6(x)} \, dx &=\frac {1}{3} \int \frac {1}{1-\cosh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1+\sqrt [3]{-1} \cosh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1-(-1)^{2/3} \cosh ^2(x)} \, dx\\ &=-\left (\frac {1}{3} \int \text {csch}^2(x) \, dx\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {1}{3} i \text {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\coth (x)}{3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 111, normalized size = 1.56 \begin {gather*} -\frac {(15+8 \cosh (2 x)+\cosh (4 x)) \sinh (x) \left (-6 \cosh (x)+\sqrt [4]{-3} \left (\left (3 i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {(-1)^{3/4} \left (-i+\sqrt {3}\right ) \tanh (x)}{2 \sqrt [4]{3}}\right )+\left (3+i \sqrt {3}\right ) \text {ArcTan}\left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (i+\sqrt {3}\right ) \tanh (x)\right )\right ) \sinh (x)\right )}{144 \left (-1+\cosh ^6(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[x]^6)^(-1),x]

[Out]

-1/144*((15 + 8*Cosh[2*x] + Cosh[4*x])*Sinh[x]*(-6*Cosh[x] + (-3)^(1/4)*((3*I + Sqrt[3])*ArcTan[((-1)^(3/4)*(-
I + Sqrt[3])*Tanh[x])/(2*3^(1/4))] + (3 + I*Sqrt[3])*ArcTan[((-1/3)^(1/4)*(I + Sqrt[3])*Tanh[x])/2])*Sinh[x]))
/(-1 + Cosh[x]^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(381\) vs. \(2(49)=98\).
time = 0.62, size = 382, normalized size = 5.38

method	result	size
risch	\(\frac {2}{3 \left ({\mathrm e}^{2 x}-1\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (3888 \textit {\_Z}^{4}-108 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-1296 \textit {\_R}^{3}+216 \textit {\_R}^{2}+{\mathrm e}^{2 x}-1\right )\right )\)	\(47\)
default	\(\frac {\tanh \left (\frac {x}{2}\right )}{6}+\frac {1}{6 \tanh \left (\frac {x}{2}\right )}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}{\tanh ^{2}\left (\frac {x}{2}\right )-\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}\right )+2 \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+1\right )+2 \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )-1\right )\right )}{72}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}{\tanh ^{2}\left (\frac {x}{2}\right )+\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}\right )+2 \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+1\right )+2 \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )-1\right )\right )}{24}+\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}{\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}-1\right )\right )}{24}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}{\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+\sqrt {3}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}-1\right )\right )}{72}\)	\(382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cosh(x)^6),x,method=_RETURNVERBOSE)

[Out]

1/6*tanh(1/2*x)+1/6/tanh(1/2*x)+1/72*3^(3/4)*2^(1/2)*(ln((tanh(1/2*x)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^
(1/2))/(tanh(1/2*x)^2-1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))+2*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)+2*
arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1))-1/24*3^(1/4)*2^(1/2)*(ln((tanh(1/2*x)^2-1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)
+1/3*3^(1/2))/(tanh(1/2*x)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))+2*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x
)+1)+2*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1))+1/24*3^(1/4)*2^(1/2)*(ln((tanh(1/2*x)^2+2^(1/2)*3^(1/4)*tanh(1/2
*x)+3^(1/2))/(tanh(1/2*x)^2-2^(1/2)*3^(1/4)*tanh(1/2*x)+3^(1/2)))+2*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1)+
2*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)-1))-1/72*3^(3/4)*2^(1/2)*(ln((tanh(1/2*x)^2-2^(1/2)*3^(1/4)*tanh(1/2*
x)+3^(1/2))/(tanh(1/2*x)^2+2^(1/2)*3^(1/4)*tanh(1/2*x)+3^(1/2)))+2*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1)+2
*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)-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)^6),x, algorithm="maxima")

[Out]

2/3/(e^(2*x) - 1) + integrate(1/3*(e^(3*x) + 4*e^(2*x) + e^x)/(e^(4*x) + 2*e^(3*x) + 6*e^(2*x) + 2*e^x + 1), x
) - integrate(1/3*(e^(3*x) - 4*e^(2*x) + e^x)/(e^(4*x) - 2*e^(3*x) + 6*e^(2*x) - 2*e^x + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (49) = 98\).
time = 0.41, size = 694, normalized size = 9.77 \begin {gather*} \frac {4 \, {\left (12^{\frac {1}{4}} \sqrt {6} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left ({\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{216} \, \sqrt {6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} + 144 \, e^{\left (2 \, x\right )} + 252} {\left ({\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 36 \, \sqrt {3} - 72\right )} + \frac {2}{3} \, \sqrt {3} {\left (2 \, \sqrt {3} - 3\right )} + \frac {1}{36} \, {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )} - {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} e^{\left (2 \, x\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {3} - 4\right ) + 4 \, {\left (12^{\frac {1}{4}} \sqrt {6} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-{\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{216} \, \sqrt {-6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} + 144 \, e^{\left (2 \, x\right )} + 252} {\left ({\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 36 \, \sqrt {3} + 72\right )} - \frac {2}{3} \, \sqrt {3} {\left (2 \, \sqrt {3} - 3\right )} + \frac {1}{36} \, {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )} - {\left (12^{\frac {3}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )}\right )} e^{\left (2 \, x\right )} + 3 \cdot 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} - 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 2 \, \sqrt {3} + 4\right ) - {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} + 144 \, e^{\left (2 \, x\right )} + 252\right ) + {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} - 12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 2\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-6 \, {\left (12^{\frac {1}{4}} \sqrt {6} {\left (\sqrt {3} + 3\right )} e^{\left (2 \, x\right )} + 12^{\frac {1}{4}} \sqrt {6} {\left (5 \, \sqrt {3} + 9\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144 \, \sqrt {3} + 36 \, e^{\left (4 \, x\right )} + 144 \, e^{\left (2 \, x\right )} + 252\right ) + 96}{144 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(x)^6),x, algorithm="fricas")

[Out]

1/144*(4*(12^(1/4)*sqrt(6)*e^(2*x) - 12^(1/4)*sqrt(6))*sqrt(-4*sqrt(3) + 8)*arctan((sqrt(3) + 2)*e^(2*x) + 1/2
16*sqrt(6*(12^(1/4)*sqrt(6)*(sqrt(3) + 3)*e^(2*x) + 12^(1/4)*sqrt(6)*(5*sqrt(3) + 9))*sqrt(-4*sqrt(3) + 8) + 1
44*sqrt(3) + 36*e^(4*x) + 144*e^(2*x) + 252)*((12^(3/4)*sqrt(6)*(sqrt(3) + 3) + 3*12^(1/4)*sqrt(6)*(sqrt(3) +
3))*sqrt(-4*sqrt(3) + 8) - 36*sqrt(3) - 72) + 2/3*sqrt(3)*(2*sqrt(3) - 3) + 1/36*(12^(3/4)*sqrt(6)*(sqrt(3) -
3) - (12^(3/4)*sqrt(6)*(sqrt(3) + 3) + 3*12^(1/4)*sqrt(6)*(sqrt(3) + 3))*e^(2*x) + 3*12^(1/4)*sqrt(6)*(sqrt(3)
- 3))*sqrt(-4*sqrt(3) + 8) + 2*sqrt(3) - 4) + 4*(12^(1/4)*sqrt(6)*e^(2*x) - 12^(1/4)*sqrt(6))*sqrt(-4*sqrt(3)
+ 8)*arctan(-(sqrt(3) + 2)*e^(2*x) + 1/216*sqrt(-6*(12^(1/4)*sqrt(6)*(sqrt(3) + 3)*e^(2*x) + 12^(1/4)*sqrt(6)
*(5*sqrt(3) + 9))*sqrt(-4*sqrt(3) + 8) + 144*sqrt(3) + 36*e^(4*x) + 144*e^(2*x) + 252)*((12^(3/4)*sqrt(6)*(sqr
t(3) + 3) + 3*12^(1/4)*sqrt(6)*(sqrt(3) + 3))*sqrt(-4*sqrt(3) + 8) + 36*sqrt(3) + 72) - 2/3*sqrt(3)*(2*sqrt(3)
- 3) + 1/36*(12^(3/4)*sqrt(6)*(sqrt(3) - 3) - (12^(3/4)*sqrt(6)*(sqrt(3) + 3) + 3*12^(1/4)*sqrt(6)*(sqrt(3) +
3))*e^(2*x) + 3*12^(1/4)*sqrt(6)*(sqrt(3) - 3))*sqrt(-4*sqrt(3) + 8) - 2*sqrt(3) + 4) - (12^(1/4)*sqrt(6)*(sq
rt(3) + 2)*e^(2*x) - 12^(1/4)*sqrt(6)*(sqrt(3) + 2))*sqrt(-4*sqrt(3) + 8)*log(6*(12^(1/4)*sqrt(6)*(sqrt(3) + 3
)*e^(2*x) + 12^(1/4)*sqrt(6)*(5*sqrt(3) + 9))*sqrt(-4*sqrt(3) + 8) + 144*sqrt(3) + 36*e^(4*x) + 144*e^(2*x) +
252) + (12^(1/4)*sqrt(6)*(sqrt(3) + 2)*e^(2*x) - 12^(1/4)*sqrt(6)*(sqrt(3) + 2))*sqrt(-4*sqrt(3) + 8)*log(-6*(
12^(1/4)*sqrt(6)*(sqrt(3) + 3)*e^(2*x) + 12^(1/4)*sqrt(6)*(5*sqrt(3) + 9))*sqrt(-4*sqrt(3) + 8) + 144*sqrt(3)
+ 36*e^(4*x) + 144*e^(2*x) + 252) + 96)/(e^(2*x) - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (65) = 130\).
time = 10.45, size = 632, normalized size = 8.90 \begin {gather*} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{24} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{72} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{72} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 4 \sqrt {3} \right )}}{24} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} - 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{24} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} - 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{72} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{72} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{24} + \frac {\tanh {\left (\frac {x}{2} \right )}}{6} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} - 1 \right )}}{12} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} - 1 \right )}}{36} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{12} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt {2} \cdot \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{36} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} - 1 \right )}}{36} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} - 1 \right )}}{12} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} + 1 \right )}}{36} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )}}{3} + 1 \right )}}{12} + \frac {1}{6 \tanh {\left (\frac {x}{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(x)**6),x)

[Out]

-sqrt(2)*3**(1/4)*log(4*tanh(x/2)**2 - 4*sqrt(2)*3**(1/4)*tanh(x/2) + 4*sqrt(3))/24 - sqrt(2)*3**(3/4)*log(4*t
anh(x/2)**2 - 4*sqrt(2)*3**(1/4)*tanh(x/2) + 4*sqrt(3))/72 + sqrt(2)*3**(3/4)*log(4*tanh(x/2)**2 + 4*sqrt(2)*3
**(1/4)*tanh(x/2) + 4*sqrt(3))/72 + sqrt(2)*3**(1/4)*log(4*tanh(x/2)**2 + 4*sqrt(2)*3**(1/4)*tanh(x/2) + 4*sqr
t(3))/24 - sqrt(2)*3**(1/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/24 - sqrt(2)*3**
(3/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/72 + sqrt(2)*3**(3/4)*log(36*tanh(x/2)
**2 + 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/72 + sqrt(2)*3**(1/4)*log(36*tanh(x/2)**2 + 12*sqrt(2)*3**(3
/4)*tanh(x/2) + 12*sqrt(3))/24 + tanh(x/2)/6 - sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) - 1)/12 + sqrt
(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) - 1)/36 - sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) + 1)/1
2 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) + 1)/36 - sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*tanh(x/2
)/3 - 1)/36 + sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 - 1)/12 - sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3
/4)*tanh(x/2)/3 + 1)/36 + sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 + 1)/12 + 1/(6*tanh(x/2))

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Giac [A]
time = 0.43, size = 10, normalized size = 0.14 \begin {gather*} \frac {2}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(x)^6),x, algorithm="giac")

[Out]

2/3/(e^(2*x) - 1)

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Mupad [B]
time = 4.52, size = 329, normalized size = 4.63 \begin {gather*} \ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}-\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}[phone]\right )+\frac {3870294016}{9}\right )+\frac {548405248}{27}\right )+\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}-\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}[phone]\right )+\frac {3870294016}{9}\right )+\frac {548405248}{27}\right )+\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}-\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}[phone]\right )+\frac {3870294016}{9}\right )+\frac {548405248}{27}\right )+\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}-\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}[phone]\right )+\frac {3870294016}{9}\right )+\frac {548405248}{27}\right )+\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\frac {2}{3\,\left ({\mathrm {e}}^{2\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cosh(x)^6 - 1),x)

[Out]

log((1061158912*exp(2*x))/27 + (1/72 - (3^(1/2)*1i)/216)^(1/2)*((2539651072*exp(2*x))/9 - (1/72 - (3^(1/2)*1i)
/216)^(1/2)*((21515730944*exp(2*x))/9 + (1/72 - (3^(1/2)*1i)/216)^(1/2)*(19788726272*exp(2*x) + 2864709632) +
3870294016/9) + 548405248/27) + 351797248/81)*(1/72 - (3^(1/2)*1i)/216)^(1/2) + log((1061158912*exp(2*x))/27 +
((3^(1/2)*1i)/216 + 1/72)^(1/2)*((2539651072*exp(2*x))/9 - ((3^(1/2)*1i)/216 + 1/72)^(1/2)*((21515730944*exp(
2*x))/9 + ((3^(1/2)*1i)/216 + 1/72)^(1/2)*(19788726272*exp(2*x) + 2864709632) + 3870294016/9) + 548405248/27)
+ 351797248/81)*((3^(1/2)*1i)/216 + 1/72)^(1/2) - log((1061158912*exp(2*x))/27 - (1/72 - (3^(1/2)*1i)/216)^(1/
2)*((2539651072*exp(2*x))/9 + (1/72 - (3^(1/2)*1i)/216)^(1/2)*((21515730944*exp(2*x))/9 - (1/72 - (3^(1/2)*1i)
/216)^(1/2)*(19788726272*exp(2*x) + 2864709632) + 3870294016/9) + 548405248/27) + 351797248/81)*(1/72 - (3^(1/
2)*1i)/216)^(1/2) - log((1061158912*exp(2*x))/27 - ((3^(1/2)*1i)/216 + 1/72)^(1/2)*((2539651072*exp(2*x))/9 +
((3^(1/2)*1i)/216 + 1/72)^(1/2)*((21515730944*exp(2*x))/9 - ((3^(1/2)*1i)/216 + 1/72)^(1/2)*(19788726272*exp(2
*x) + 2864709632) + 3870294016/9) + 548405248/27) + 351797248/81)*((3^(1/2)*1i)/216 + 1/72)^(1/2) + 2/(3*(exp(
2*x) - 1))

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