### 3.1029 $$\int \text{csch}(x) \sqrt{1+\log ^2(\coth (x))} \text{sech}(x) \, dx$$

Optimal. Leaf size=27 $-\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x)))$

[Out]

-ArcSinh[Log[Coth[x]]]/2 - (Log[Coth[x]]*Sqrt[1 + Log[Coth[x]]^2])/2

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Rubi [A]  time = 0.17014, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {6696, 195, 215} $-\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x)))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[x]*Sqrt[1 + Log[Coth[x]]^2]*Sech[x],x]

[Out]

-ArcSinh[Log[Coth[x]]]/2 - (Log[Coth[x]]*Sqrt[1 + Log[Coth[x]]^2])/2

Rule 6696

Int[(u_.)*((a_.) + (b_.)*(y_)^(n_))^(p_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[In
t[(a + b*x^n)^p, x], x, y], x] /;  !FalseQ[q]] /; FreeQ[{a, b, n, p}, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.033826, size = 27, normalized size = 1. $-\frac{1}{2} \log (\coth (x)) \sqrt{\log ^2(\coth (x))+1}-\frac{1}{2} \sinh ^{-1}(\log (\coth (x)))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[x]*Sqrt[1 + Log[Coth[x]]^2]*Sech[x],x]

[Out]

-ArcSinh[Log[Coth[x]]]/2 - (Log[Coth[x]]*Sqrt[1 + Log[Coth[x]]^2])/2

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Maple [A]  time = 0.074, size = 22, normalized size = 0.8 \begin{align*} -{\frac{{\it Arcsinh} \left ( \ln \left ({\rm coth} \left (x\right ) \right ) \right ) }{2}}-{\frac{\ln \left ({\rm coth} \left (x\right ) \right ) }{2}\sqrt{1+ \left ( \ln \left ({\rm coth} \left (x\right ) \right ) \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arcsinh(ln(coth(x)))-1/2*ln(coth(x))*(1+ln(coth(x))^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\log \left (\coth \left (x\right )\right )^{2} + 1} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(log(coth(x))^2 + 1)*csch(x)*sech(x), x)

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Fricas [B]  time = 2.22986, size = 171, normalized size = 6.33 \begin{align*} -\frac{1}{2} \, \sqrt{\log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )^{2} + 1} \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + \frac{1}{2} \, \log \left (\sqrt{\log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )^{2} + 1} - \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(log(cosh(x)/sinh(x))^2 + 1)*log(cosh(x)/sinh(x)) + 1/2*log(sqrt(log(cosh(x)/sinh(x))^2 + 1) - log(co
sh(x)/sinh(x)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\log \left (\coth \left (x\right )\right )^{2} + 1} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(log(coth(x))^2 + 1)*csch(x)*sech(x), x)