3.11.29 \(\int \text {csch}(x) \sqrt {1+\log ^2(\coth (x))} \text {sech}(x) \, dx\) [1029]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {6828, 201, 221} \begin {gather*} -\frac {1}{2} \log (\coth (x)) \sqrt {\log ^2(\coth (x))+1}-\frac {1}{2} \sinh ^{-1}(\log (\coth (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 201

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 221

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]

Rule 6828

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]

Rubi steps

\begin {align*} \int \text {csch}(x) \sqrt {1+\log ^2(\coth (x))} \text {sech}(x) \, dx &=-\text {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} \text {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*}

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Mathematica [A]
time = 0.03, size = 39, normalized size = 1.44 \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\frac {\log (\coth (x))}{\sqrt {1+\log ^2(\coth (x))}}\right )-\frac {1}{2} \log (\coth (x)) \sqrt {1+\log ^2(\coth (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 5.47, size = 22, normalized size = 0.81

method	result	size
derivativedivides	\(-\frac {\arcsinh \left (\ln \left (\coth \left (x \right )\right )\right )}{2}-\frac {\ln \left (\coth \left (x \right )\right ) \sqrt {1+\ln \left (\coth \left (x \right )\right )^{2}}}{2}\)	\(22\)
default	\(-\frac {\arcsinh \left (\ln \left (\coth \left (x \right )\right )\right )}{2}-\frac {\ln \left (\coth \left (x \right )\right ) \sqrt {1+\ln \left (\coth \left (x \right )\right )^{2}}}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).
time = 0.35, size = 53, normalized size = 1.96 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 1.88, size = 21, normalized size = 0.78 \begin {gather*} -\frac {\mathrm {asinh}\left (\ln \left (\mathrm {coth}\left (x\right )\right )\right )}{2}-\frac {\ln \left (\mathrm {coth}\left (x\right )\right )\,\sqrt {{\ln \left (\mathrm {coth}\left (x\right )\right )}^2+1}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(coth(x))^2 + 1)^(1/2)/(cosh(x)*sinh(x)),x)

[Out]

- asinh(log(coth(x)))/2 - (log(coth(x))*(log(coth(x))^2 + 1)^(1/2))/2

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