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3.74 \(\int \frac{1}{1-\cosh ^6(x)} \, dx\)

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]

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

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Rubi [A]  time = 0.122465, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3211, 3181, 206, 3175, 3767, 8} \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+\sqrt [3]{-1}}}\right )}{3 \sqrt{1+\sqrt [3]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-(-1)^{2/3}}}\right )}{3 \sqrt{1-(-1)^{2/3}}}+\frac{\coth (x)}{3} \]

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 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])

Rule 3175

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 3767

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]

Rule 8

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

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} \operatorname{Subst}\left (\int \frac{1}{1-\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\coth (x)\right )+\frac{1}{3} \operatorname{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 \operatorname{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*}

Mathematica [C]  time = 0.219157, size = 111, normalized size = 1.56 \[ -\frac{\sinh (x) (8 \cosh (2 x)+\cosh (4 x)+15) \left (-6 \cosh (x)+\sqrt [4]{-3} \sinh (x) \left (\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{(-1)^{3/4} \left (\sqrt{3}-i\right ) \tanh (x)}{2 \sqrt [4]{3}}\right )+\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{1}{2} \sqrt [4]{-\frac{1}{3}} \left (\sqrt{3}+i\right ) \tanh (x)\right )\right )\right )}{144 \left (\cosh ^6(x)-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[x]^6)^(-1),x]

[Out]

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

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Maple [B]  time = 0.023, size = 426, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cosh(x)^6),x)

[Out]

1/6*tanh(1/2*x)+1/36*3^(3/4)*2^(1/2)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)+1/36*3^(3/4)*2^(1/2)*arctan(2^(1/2)
*3^(1/4)*tanh(1/2*x)-1)+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))/(t
anh(1/2*x)^2-1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))-1/24*2^(1/2)*3^(1/4)*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)))-1/12*2^(1/2)*3^(
1/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)-1/12*2^(1/2)*3^(1/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1)+1/6/tanh
(1/2*x)+1/12*3^(1/4)*2^(1/2)*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1)+1/12*3^(1/4)*2^(1/2)*arctan(1/3*3^(3/4)
*tanh(1/2*x)*2^(1/2)-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)))-1/36*3^(3/4)*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)))-1/36*3^(3/4)*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., size = 0, normalized size = 0. \begin{align*} \frac{2}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} + \int \frac{e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + e^{x}}{3 \,{\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )}}\,{d x} - \int \frac{e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + e^{x}}{3 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}\,{d x} \end{align*}

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]  time = 2.54127, size = 2302, normalized size = 32.42 \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)^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 [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)**6),x)

[Out]

Timed out

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Giac [A]  time = 1.20162, size = 14, normalized size = 0.2 \begin{align*} \frac{2}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}

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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