3.425 \(\int \frac{\coth (x)}{1-\sinh ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ \log (\sinh (x))-\frac{1}{2} \log \left (1-\sinh ^2(x)\right ) \]

[Out]

Log[Sinh[x]] - Log[1 - Sinh[x]^2]/2

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Rubi [A]  time = 0.0324463, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3194, 36, 31, 29} \[ \log (\sinh (x))-\frac{1}{2} \log \left (1-\sinh ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(1 - Sinh[x]^2),x]

[Out]

Log[Sinh[x]] - Log[1 - Sinh[x]^2]/2

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.013551, size = 23, normalized size = 1.35 \[ -2 \left (\frac{1}{4} \log \left (1-\sinh ^2(x)\right )-\frac{1}{2} \log (\sinh (x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(1 - Sinh[x]^2),x]

[Out]

-2*(-Log[Sinh[x]]/2 + Log[1 - Sinh[x]^2]/4)

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Maple [B]  time = 0.029, size = 41, normalized size = 2.4 \begin{align*} -{\frac{1}{2}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) }-{\frac{1}{2}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) }+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.109, size = 61, normalized size = 3.59 \begin{align*} -\frac{1}{2} \, \log \left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \frac{1}{2} \, \log \left (-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.78169, size = 165, normalized size = 9.71 \begin{align*} -\frac{1}{2} \, \log \left (\frac{2 \,{\left (\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(2*(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + log(2*sinh(x)/(cosh(x) -
sinh(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1704, size = 34, normalized size = 2. \begin{align*} -\frac{1}{2} \, \log \left ({\left | e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(e^(4*x) - 6*e^(2*x) + 1)) + log(abs(e^(2*x) - 1))