3.672 \(\int \frac {1}{(-\coth (x)+\text {csch}(x))^4} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac {2 \sinh (x)}{1-\cosh (x)} \]

[Out]

x+2*sinh(x)/(1-cosh(x))+2/3*sinh(x)^3/(1-cosh(x))^3

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4392, 2680, 8} \[ x+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac {2 \sinh (x)}{1-\cosh (x)} \]

Antiderivative was successfully verified.

[In]

Int[(-Coth[x] + Csch[x])^(-4),x]

[Out]

x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)

Rule 8

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

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^4} \, dx &=\int \frac {\sinh ^4(x)}{(i-i \cosh (x))^4} \, dx\\ &=\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}-\int \frac {\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\int 1 \, dx\\ &=x+\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 28, normalized size = 0.93 \[ -\frac {2}{3} \coth ^3\left (\frac {x}{2}\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2\left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Coth[x] + Csch[x])^(-4),x]

[Out]

(-2*Coth[x/2]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[x/2]^2])/3

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 68, normalized size = 2.27 \[ \frac {3 \, x \cosh \relax (x)^{2} + 3 \, x \sinh \relax (x)^{2} - 4 \, {\left (3 \, x + 10\right )} \cosh \relax (x) + 2 \, {\left (3 \, x \cosh \relax (x) - 3 \, x - 4\right )} \sinh \relax (x) + 9 \, x + 24}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 4 \, \cosh \relax (x) + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))^4,x, algorithm="fricas")

[Out]

1/3*(3*x*cosh(x)^2 + 3*x*sinh(x)^2 - 4*(3*x + 10)*cosh(x) + 2*(3*x*cosh(x) - 3*x - 4)*sinh(x) + 9*x + 24)/(cos
h(x)^2 + 2*(cosh(x) - 1)*sinh(x) + sinh(x)^2 - 4*cosh(x) + 3)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 22, normalized size = 0.73 \[ x - \frac {8 \, {\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 2\right )}}{3 \, {\left (e^{x} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))^4,x, algorithm="giac")

[Out]

x - 8/3*(3*e^(2*x) - 3*e^x + 2)/(e^x - 1)^3

________________________________________________________________________________________

maple [A]  time = 0.22, size = 34, normalized size = 1.13 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {2}{\tanh \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-coth(x)+csch(x))^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 38, normalized size = 1.27 \[ x - \frac {8 \, {\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))^4,x, algorithm="maxima")

[Out]

x - 8/3*(3*e^(-x) - 3*e^(-2*x) - 2)/(3*e^(-x) - 3*e^(-2*x) + e^(-3*x) - 1)

________________________________________________________________________________________

mupad [B]  time = 1.53, size = 59, normalized size = 1.97 \[ x-\frac {8\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {8}{3}}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}-\frac {8}{3\,\left ({\mathrm {e}}^x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) - 1/sinh(x))^4,x)

[Out]

x - (8*exp(x))/(3*(exp(2*x) - 2*exp(x) + 1)) + ((8*exp(2*x))/3 + 8/3)/(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1) -
8/(3*(exp(x) - 1))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x))**4,x)

[Out]

Integral((-coth(x) + csch(x))**(-4), x)

________________________________________________________________________________________