### 3.657 $$\int (\coth (x)+\text{csch}(x))^2 \, dx$$

Optimal. Leaf size=14 $x+\frac{2 \sinh (x)}{1-\cosh (x)}$

[Out]

x + (2*Sinh[x])/(1 - Cosh[x])

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Rubi [A]  time = 0.0822005, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.571, Rules used = {4392, 2670, 2680, 8} $x+\frac{2 \sinh (x)}{1-\cosh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^2,x]

[Out]

x + (2*Sinh[x])/(1 - Cosh[x])

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]

Rule 2670

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a/g)^
(2*m), Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 -
b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]

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 8

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

Rubi steps

\begin{align*} \int (\coth (x)+\text{csch}(x))^2 \, dx &=-\int (i+i \cosh (x))^2 \text{csch}^2(x) \, 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*}

Mathematica [A]  time = 0.0258454, size = 10, normalized size = 0.71 $x-2 \coth \left (\frac{x}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^2,x]

[Out]

x - 2*Coth[x/2]

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Maple [A]  time = 0.009, size = 21, normalized size = 1.5 \begin{align*} x-2\,{\rm coth} \left (x\right )-2\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\sinh \left ( x \right ) }}+2\,\sinh \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-2*coth(x)-2*cosh(x)^2/sinh(x)+2*sinh(x)

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Maxima [B]  time = 1.04969, size = 34, normalized size = 2.43 \begin{align*} x + \frac{4}{e^{\left (-x\right )} - e^{x}} + \frac{4}{e^{\left (-2 \, x\right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4/(e^(-x) - e^x) + 4/(e^(-2*x) - 1)

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Fricas [A]  time = 1.96929, size = 77, normalized size = 5.5 \begin{align*} \frac{x \cosh \left (x\right ) + x \sinh \left (x\right ) - x - 4}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x) + csch(x))**2, x)

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Giac [A]  time = 1.10915, size = 14, normalized size = 1. \begin{align*} x - \frac{4}{e^{x} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 4/(e^x - 1)