3.94 \(\int \frac{\text{csch}^2(x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=7 \[ -\log (\coth (x)+1) \]

[Out]

-Log[1 + Coth[x]]

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Rubi [A]  time = 0.0357077, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3487, 31} \[ -\log (\coth (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(1 + Coth[x]),x]

[Out]

-Log[1 + Coth[x]]

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.0029622, size = 7, normalized size = 1. \[ \log (\sinh (x))-x \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2/(1 + Coth[x]),x]

[Out]

-x + Log[Sinh[x]]

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Maple [A]  time = 0.017, size = 8, normalized size = 1.1 \begin{align*} -\ln \left ( 1+{\rm coth} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(1+coth(x))

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Maxima [A]  time = 1.05112, size = 9, normalized size = 1.29 \begin{align*} -\log \left (\coth \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(coth(x) + 1)

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Fricas [B]  time = 2.59185, size = 58, normalized size = 8.29 \begin{align*} -2 \, x + \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(csch(x)^2/(1+coth(x)),x, algorithm="fricas")

[Out]

-2*x + 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{\operatorname{csch}^{2}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)**2/(coth(x) + 1), x)

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Giac [A]  time = 1.15829, size = 16, normalized size = 2.29 \begin{align*} -2 \, x + \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(csch(x)^2/(1+coth(x)),x, algorithm="giac")

[Out]

-2*x + log(abs(e^(2*x) - 1))