3.59 \(\int \frac{1}{1-\cosh ^3(x)} \, dx\)

Optimal. Leaf size=95 \[ -\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))} \]

[Out]

(-2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 - (-1)^(2/3))) - (2*(-1)^(1/4)*ArcTanh[((-1
)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 + (-1)^(1/3))) - Sinh[x]/(3*(1 - Cosh[x]))

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Rubi [A]  time = 0.121, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3213, 2648, 2659, 208, 205} \[ -\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cosh[x]^3)^(-1),x]

[Out]

(-2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 - (-1)^(2/3))) - (2*(-1)^(1/4)*ArcTanh[((-1
)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 + (-1)^(1/3))) - Sinh[x]/(3*(1 - Cosh[x]))

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]))

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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{1-\cosh ^3(x)} \, dx &=\int \left (\frac{1}{3 (1-\cosh (x))}+\frac{1}{3 \left (1+\sqrt [3]{-1} \cosh (x)\right )}+\frac{1}{3 \left (1-(-1)^{2/3} \cosh (x)\right )}\right ) \, dx\\ &=\frac{1}{3} \int \frac{1}{1-\cosh (x)} \, dx+\frac{1}{3} \int \frac{1}{1+\sqrt [3]{-1} \cosh (x)} \, dx+\frac{1}{3} \int \frac{1}{1-(-1)^{2/3} \cosh (x)} \, dx\\ &=-\frac{\sinh (x)}{3 (1-\cosh (x))}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{-1}-\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-(-1)^{2/3}-\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\frac{2 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1-(-1)^{2/3}\right )}-\frac{2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac{(-1)^{3/4} \tanh \left (\frac{x}{2}\right )}{\sqrt [4]{3}}\right )}{3^{3/4} \left (1+\sqrt [3]{-1}\right )}-\frac{\sinh (x)}{3 (1-\cosh (x))}\\ \end{align*}

Mathematica [C]  time = 0.547871, size = 147, normalized size = 1.55 \[ \frac{1}{3} \coth \left (\frac{x}{2}\right )+\frac{\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{\left (1-i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{2 \left (3-i \sqrt{3}\right )}}\right )}{3 \sqrt{\frac{3}{2} \left (3-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{\left (1+i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{2 \left (3+i \sqrt{3}\right )}}\right )}{3 \sqrt{\frac{3}{2} \left (3+i \sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[x]^3)^(-1),x]

[Out]

((3*I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*Tanh[x/2])/Sqrt[2*(3 - I*Sqrt[3])]])/(3*Sqrt[(3*(3 - I*Sqrt[3]))/2])
+ ((-3*I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*Tanh[x/2])/Sqrt[2*(3 + I*Sqrt[3])]])/(3*Sqrt[(3*(3 + I*Sqrt[3]))/2
]) + Coth[x/2]/3

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Maple [B]  time = 0.016, size = 212, normalized size = 2.2 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{\sqrt [4]{3}\sqrt{2}}{6}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+1 \right ) }+{\frac{\sqrt [4]{3}\sqrt{2}}{6}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }-1 \right ) }+{\frac{\sqrt [4]{3}\sqrt{2}}{12}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) ^{-1}} \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{36}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +\sqrt{3} \right ) ^{-1}} \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+1 \right ) }-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ({\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cosh(x)^3),x)

[Out]

1/3/tanh(1/2*x)+1/6*3^(1/4)*2^(1/2)*arctan(1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1)+1/6*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)*ln((tanh(1/2*x)^2+2^(1/2)*3^(1/4)*tanh(1/2*x)+3^(1/2))/(tan
h(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)*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)))-1/18*3^(3/4)*2^(1/2)*arctan(1/3*3^(3/4)*ta
nh(1/2*x)*2^(1/2)+1)-1/18*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^{x} - 1\right )}} + \int \frac{2 \,{\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{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)^3),x, algorithm="maxima")

[Out]

2/3/(e^x - 1) + integrate(2/3*(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 = 2.50265, size = 1924, normalized size = 20.25 \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)^3),x, algorithm="fricas")

[Out]

-1/36*(4*(3^(3/4)*e^x - 3^(3/4))*sqrt(-4*sqrt(3) + 8)*arctan(1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x
- 1/48*(2*sqrt(3)*(sqrt(3) + 3) - (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6
*sqrt(3) + 18)*sqrt(2*(3^(1/4)*(sqrt(3) + 2) + 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) + 4*e
^x + 4) - 1/12*sqrt(3)*(sqrt(3) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x + 3^(3/4)
*(sqrt(3) + 1) + 3*3^(1/4)*(sqrt(3) - 1))*sqrt(-4*sqrt(3) + 8) - 1/4*sqrt(3) + 1/4) + 4*(3^(3/4)*e^x - 3^(3/4)
)*sqrt(-4*sqrt(3) + 8)*arctan(-1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x + 1/48*(2*sqrt(3)*(sqrt(3) + 3
) + (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6*sqrt(3) + 18)*sqrt(-2*(3^(1/4
)*(sqrt(3) + 2) + 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) + 4*e^x + 4) + 1/12*sqrt(3)*(sqrt(
3) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x + 3^(3/4)*(sqrt(3) + 1) + 3*3^(1/4)*(s
qrt(3) - 1))*sqrt(-4*sqrt(3) + 8) + 1/4*sqrt(3) - 1/4) + (3^(1/4)*(2*sqrt(3) + 3)*e^x - 3^(1/4)*(2*sqrt(3) + 3
))*sqrt(-4*sqrt(3) + 8)*log(2*(3^(1/4)*(sqrt(3) + 2) + 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*
x) + 4*e^x + 4) - (3^(1/4)*(2*sqrt(3) + 3)*e^x - 3^(1/4)*(2*sqrt(3) + 3))*sqrt(-4*sqrt(3) + 8)*log(-2*(3^(1/4)
*(sqrt(3) + 2) + 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) + 4*e^x + 4) - 24)/(e^x - 1)

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Sympy [B]  time = 5.28013, size = 405, normalized size = 4.26 \begin{align*} \frac{\sqrt{2} \sqrt [4]{3} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} - 4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} + 4 \sqrt{3} \right )} \tanh{\left (\frac{x}{2} \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} + 4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} + 4 \sqrt{3} \right )} \tanh{\left (\frac{x}{2} \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} - 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} - 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{4 \sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} + 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )}}{3} + 1 \right )}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} - \frac{6}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} + \frac{2 \sqrt{3}}{- 18 \tanh{\left (\frac{x}{2} \right )} + 6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(x)**3),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))*tanh(x/2)/(-18*tanh(x/2) + 6*s
qrt(3)*tanh(x/2)) - sqrt(2)*3**(1/4)*log(4*tanh(x/2)**2 + 4*sqrt(2)*3**(1/4)*tanh(x/2) + 4*sqrt(3))*tanh(x/2)/
(-18*tanh(x/2) + 6*sqrt(3)*tanh(x/2)) - 4*sqrt(2)*3**(1/4)*tanh(x/2)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 - 1)/(-
18*tanh(x/2) + 6*sqrt(3)*tanh(x/2)) + 2*sqrt(2)*3**(3/4)*tanh(x/2)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 - 1)/(-18
*tanh(x/2) + 6*sqrt(3)*tanh(x/2)) - 4*sqrt(2)*3**(1/4)*tanh(x/2)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 + 1)/(-18*t
anh(x/2) + 6*sqrt(3)*tanh(x/2)) + 2*sqrt(2)*3**(3/4)*tanh(x/2)*atan(sqrt(2)*3**(3/4)*tanh(x/2)/3 + 1)/(-18*tan
h(x/2) + 6*sqrt(3)*tanh(x/2)) - 6/(-18*tanh(x/2) + 6*sqrt(3)*tanh(x/2)) + 2*sqrt(3)/(-18*tanh(x/2) + 6*sqrt(3)
*tanh(x/2))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(x)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError