### 3.1022 $$\int (-1-\frac{1}{1-\coth ^2(x)}) \text{csch}^2(x) \, dx$$

Optimal. Leaf size=4 $x+\coth (x)$

[Out]

x + Coth[x]

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Rubi [A]  time = 0.0618662, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {453, 206} $x+\coth (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(-1 - (1 - Coth[x]^2)^(-1))*Csch[x]^2,x]

[Out]

x + Coth[x]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (-1-\frac{1}{1-\coth ^2(x)}\right ) \text{csch}^2(x) \, dx &=-\operatorname{Subst}\left (\int \frac{1-2 x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\coth (x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x+\coth (x)\\ \end{align*}

Mathematica [A]  time = 0.0057326, size = 4, normalized size = 1. $x+\coth (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-1 - (1 - Coth[x]^2)^(-1))*Csch[x]^2,x]

[Out]

x + Coth[x]

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Maple [B]  time = 0.03, size = 32, normalized size = 8. \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-1-1/(1-coth(x)^2))*csch(x)^2,x)

[Out]

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

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Maxima [B]  time = 1.06225, size = 16, normalized size = 4. \begin{align*} x - \frac{2}{e^{\left (-2 \, x\right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2/(e^(-2*x) - 1)

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Fricas [B]  time = 1.97203, size = 50, normalized size = 12.5 \begin{align*} \frac{{\left (x - 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}{\sinh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x - 1)*sinh(x) + cosh(x))/sinh(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{2 \operatorname{csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\, dx - \int \frac{\coth ^{2}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-2*csch(x)**2/(coth(x)**2 - 1), x) - Integral(coth(x)**2*csch(x)**2/(coth(x)**2 - 1), x)

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Giac [B]  time = 1.11284, size = 16, normalized size = 4. \begin{align*} x + \frac{2}{e^{\left (2 \, x\right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2/(e^(2*x) - 1)