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

Optimal. Leaf size=26 \[ x-\frac {2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac {2 \sinh (x)}{\cosh (x)+1} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4392, 2680, 8} \[ x-\frac {2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac {2 \sinh (x)}{\cosh (x)+1} \]

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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.15 \[ x-\frac {8}{3} \tanh \left (\frac {x}{2}\right )+\frac {2}{3} \tanh \left (\frac {x}{2}\right ) \text {sech}^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^(-4),x]

[Out]

x - (8*Tanh[x/2])/3 + (2*Sech[x/2]^2*Tanh[x/2])/3

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fricas [B]  time = 0.41, size = 68, normalized size = 2.62 \[ \frac {3 \, x \cosh \relax (x)^{2} + 3 \, x \sinh \relax (x)^{2} + 4 \, {\left (3 \, x + 10\right )} \cosh \relax (x) + 2 \, {\left (3 \, x \cosh \relax (x) + 3 \, x + 4\right )} \sinh \relax (x) + 9 \, x + 24}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )}} \]

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)

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giac [A]  time = 0.12, size = 22, normalized size = 0.85 \[ x + \frac {8 \, {\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]

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

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maple [A]  time = 0.21, size = 32, normalized size = 1.23 \[ -\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.66, size = 38, normalized size = 1.46 \[ x - \frac {8 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + 2\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \]

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)

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mupad [B]  time = 1.55, size = 57, normalized size = 2.19 \[ x+\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {8}{3}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {8\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}+\frac {8}{3\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1/sinh(x))^4,x)

[Out]

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

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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)

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