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 = 0.06, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ x+\coth (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + Coth[x]

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fricas [B]  time = 0.45, size = 14, normalized size = 3.50 \[ \frac {{\left (x - 1\right )} \sinh \relax (x) + \cosh \relax (x)}{\sinh \relax (x)} \]

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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giac [B]  time = 0.13, size = 12, normalized size = 3.00 \[ x + \frac {2}{e^{\left (2 \, x\right )} - 1} \]

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)

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maple [B]  time = 0.17, size = 32, normalized size = 8.00 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{2 \tanh \left (\frac {x}{2}\right )} \]

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

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maxima [B]  time = 0.31, size = 12, normalized size = 3.00 \[ x - \frac {2}{e^{\left (-2 \, x\right )} - 1} \]

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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mupad [B]  time = 1.83, size = 12, normalized size = 3.00 \[ x+\frac {2}{{\mathrm {e}}^{2\,x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(coth(x)^2 - 1) - 1)/sinh(x)^2,x)

[Out]

x + 2/(exp(2*x) - 1)

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

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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