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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{4} \log ^2(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Tanh[x]]^2/4

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fricas [A]  time = 0.42, size = 12, normalized size = 1.33 \[ \frac {1}{4} \, \log \left (\frac {\sinh \relax (x)}{\cosh \relax (x)}\right )^{2} \]

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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giac [B]  time = 0.13, size = 20, normalized size = 2.22 \[ \frac {1}{4} \, \log \left (\frac {e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tanh \relax (x )\right )^{2}}{4} \]

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 = 0.30, size = 7, normalized size = 0.78 \[ \frac {1}{4} \, \log \left (\tanh \relax (x)\right )^{2} \]

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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mupad [B]  time = 1.67, size = 21, normalized size = 2.33 \[ \frac {{\left (\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\ln \left ({\mathrm {e}}^{2\,x}+1\right )\right )}^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(tanh(x))/sinh(2*x),x)

[Out]

(log(exp(2*x) - 1) - log(exp(2*x) + 1))^2/4

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\tanh {\relax (x )} \right )} \operatorname {csch}{\left (2 x \right )}\, dx \]

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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