### 3.1012 $$\int \text{csch}(x) \log (\tanh (x)) \text{sech}(x) \, dx$$

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

[Out]

Log[Tanh[x]]^2/2

________________________________________________________________________________________

Rubi [A]  time = 0.0260572, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {2620, 29, 6686} $\frac{1}{2} \log ^2(\tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[Tanh[x]]^2/2

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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}(x) \log (\tanh (x)) \text{sech}(x) \, dx &=\frac{1}{2} \log ^2(\tanh (x))\\ \end{align*}

Mathematica [A]  time = 0.0058762, size = 9, normalized size = 1. $\frac{1}{2} \log ^2(\tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[Tanh[x]]^2/2

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tanh \left ( x \right ) \right ) \right ) ^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 4.90216, size = 128, normalized size = 14.22 \begin{align*}{\left (\log \left (e^{x} + 1\right ) + \log \left (-e^{x} + 1\right )\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right )^{2} - \frac{1}{2} \, \log \left (e^{x} + 1\right )^{2} - \log \left (e^{x} + 1\right ) \log \left (-e^{x} + 1\right ) - \frac{1}{2} \, \log \left (-e^{x} + 1\right )^{2} +{\left (\log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )} \log \left (\tanh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.98701, size = 38, normalized size = 4.22 \begin{align*} \frac{1}{2} \, \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(x)*log(tanh(x))*sech(x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(tanh(x))*csch(x)*sech(x), x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(x)*log(tanh(x))*sech(x), x)