3.660 \(\int \frac {1}{(\coth (x)+\text {csch}(x))^2} \, dx\)

Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4392, 2680, 8} \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(Coth[x] + Csch[x])^(-2),x]

[Out]

x - (2*Sinh[x])/(1 + Cosh[x])

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))^2} \, dx &=-\int \frac {\sinh ^2(x)}{(i+i \cosh (x))^2} \, dx\\ &=-\frac {2 \sinh (x)}{1+\cosh (x)}+\int 1 \, dx\\ &=x-\frac {2 \sinh (x)}{1+\cosh (x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 10, normalized size = 0.83 \[ x-2 \tanh \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^(-2),x]

[Out]

x - 2*Tanh[x/2]

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fricas [A]  time = 0.40, size = 20, normalized size = 1.67 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + x + 4}{\cosh \relax (x) + \sinh \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^2,x, algorithm="fricas")

[Out]

(x*cosh(x) + x*sinh(x) + x + 4)/(cosh(x) + sinh(x) + 1)

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giac [A]  time = 0.13, size = 10, normalized size = 0.83 \[ x + \frac {4}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^2,x, algorithm="giac")

[Out]

x + 4/(e^x + 1)

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maple [A]  time = 0.19, size = 24, normalized size = 2.00 \[ -2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)+csch(x))^2,x)

[Out]

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

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maxima [A]  time = 0.47, size = 12, normalized size = 1.00 \[ x - \frac {4}{e^{\left (-x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^2,x, algorithm="maxima")

[Out]

x - 4/(e^(-x) + 1)

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mupad [B]  time = 0.18, size = 10, normalized size = 0.83 \[ x+\frac {4}{{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1/sinh(x))^2,x)

[Out]

x + 4/(exp(x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))**2,x)

[Out]

Integral((coth(x) + csch(x))**(-2), x)

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