### 3.1013 $$\int \text{csch}(2 x) \log (\tanh (x)) \, dx$$

Optimal. Leaf size=9 $\frac{1}{4} \log ^2(\tanh (x))$

[Out]

Log[Tanh[x]]^2/4

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Rubi [A]  time = 0.0243586, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3770, 6686} $\frac{1}{4} \log ^2(\tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[2*x]*Log[Tanh[x]],x]

[Out]

Log[Tanh[x]]^2/4

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \text{csch}(2 x) \log (\tanh (x)) \, dx &=\frac{1}{4} \log ^2(\tanh (x))\\ \end{align*}

Mathematica [A]  time = 0.0090192, size = 9, normalized size = 1. $\frac{1}{4} \log ^2(\tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[2*x]*Log[Tanh[x]],x]

[Out]

Log[Tanh[x]]^2/4

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Maple [A]  time = 0.015, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tanh \left ( x \right ) \right ) \right ) ^{2}}{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(2*x)*ln(tanh(x)),x)

[Out]

1/4*ln(tanh(x))^2

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Maxima [A]  time = 1.01263, size = 9, normalized size = 1. \begin{align*} \frac{1}{4} \, \log \left (\tanh \left (x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*log(tanh(x)),x, algorithm="maxima")

[Out]

1/4*log(tanh(x))^2

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Fricas [A]  time = 2.11135, size = 38, normalized size = 4.22 \begin{align*} \frac{1}{4} \, \log \left (\frac{\sinh \left (x\right )}{\cosh \left (x\right )}\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*log(tanh(x)),x, algorithm="fricas")

[Out]

1/4*log(sinh(x)/cosh(x))^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\tanh{\left (x \right )} \right )} \operatorname{csch}{\left (2 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*ln(tanh(x)),x)

[Out]

Integral(log(tanh(x))*csch(2*x), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*x)*log(tanh(x)),x, algorithm="giac")

[Out]

integrate(csch(2*x)*log(tanh(x)), x)