3.215 \(\int \log (a \text {sech}(x)) \, dx\)

Optimal. Leaf size=38 \[ x \log (a \text {sech}(x))+\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right )-\frac {x^2}{2}+x \log \left (e^{2 x}+1\right ) \]

[Out]

-1/2*x^2+x*ln(1+exp(2*x))+x*ln(a*sech(x))+1/2*polylog(2,-exp(2*x))

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2548, 3718, 2190, 2279, 2391} \[ \frac {1}{2} \text {PolyLog}\left (2,-e^{2 x}\right )+x \log (a \text {sech}(x))-\frac {x^2}{2}+x \log \left (e^{2 x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[a*Sech[x]],x]

[Out]

-x^2/2 + x*Log[1 + E^(2*x)] + x*Log[a*Sech[x]] + PolyLog[2, -E^(2*x)]/2

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \log (a \text {sech}(x)) \, dx &=x \log (a \text {sech}(x))+\int x \tanh (x) \, dx\\ &=-\frac {x^2}{2}+x \log (a \text {sech}(x))+2 \int \frac {e^{2 x} x}{1+e^{2 x}} \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))-\int \log \left (1+e^{2 x}\right ) \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))+\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 37, normalized size = 0.97 \[ \frac {1}{2} \left (x \left (2 \log (a \text {sech}(x))+x+2 \log \left (e^{-2 x}+1\right )\right )-\text {Li}_2\left (-e^{-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a*Sech[x]],x]

[Out]

(x*(x + 2*Log[1 + E^(-2*x)] + 2*Log[a*Sech[x]]) - PolyLog[2, -E^(-2*x)])/2

________________________________________________________________________________________

fricas [C]  time = 0.50, size = 84, normalized size = 2.21 \[ -\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right ) + x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) + x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) + {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) + {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*sech(x)),x, algorithm="fricas")

[Out]

-1/2*x^2 + x*log(2*(a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)) + x*log(I*cosh(x)
+ I*sinh(x) + 1) + x*log(-I*cosh(x) - I*sinh(x) + 1) + dilog(I*cosh(x) + I*sinh(x)) + dilog(-I*cosh(x) - I*sin
h(x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (a \operatorname {sech}\relax (x)\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*sech(x)), x)

________________________________________________________________________________________

maple [C]  time = 1.15, size = 314, normalized size = 8.26 \[ -\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{3}}{2}-\frac {x^{2}}{2}+x \ln \relax (a )+x \ln \left ({\mathrm e}^{x}\right )+\ln \left (-i {\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}\right )+\ln \left (i {\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{2 x}+1\right ) \ln \left ({\mathrm e}^{x}\right )+\ln \relax (2) x +\dilog \left (-i {\mathrm e}^{x}+1\right )+\dilog \left (i {\mathrm e}^{x}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*sech(x)),x)

[Out]

x*ln(exp(x))+1/2*I*Pi*csgn(I/(exp(2*x)+1))*csgn(I*exp(x)/(exp(2*x)+1))^2*x-1/2*I*Pi*csgn(I*a)*csgn(I*exp(x)/(e
xp(2*x)+1))*csgn(I*a/(exp(2*x)+1)*exp(x))*x+1/2*I*Pi*csgn(I*exp(x))*csgn(I*exp(x)/(exp(2*x)+1))^2*x-1/2*I*Pi*c
sgn(I*exp(x))*csgn(I/(exp(2*x)+1))*csgn(I*exp(x)/(exp(2*x)+1))*x-1/2*I*Pi*csgn(I*a/(exp(2*x)+1)*exp(x))^3*x+1/
2*I*Pi*csgn(I*a)*csgn(I*a/(exp(2*x)+1)*exp(x))^2*x+ln(2)*x+x*ln(a)-1/2*x^2+1/2*I*Pi*csgn(I*exp(x)/(exp(2*x)+1)
)*csgn(I*a/(exp(2*x)+1)*exp(x))^2*x-1/2*I*Pi*csgn(I*exp(x)/(exp(2*x)+1))^3*x-ln(exp(2*x)+1)*ln(exp(x))+ln(I*ex
p(x)+1)*ln(exp(x))+ln(-I*exp(x)+1)*ln(exp(x))+dilog(I*exp(x)+1)+dilog(-I*exp(x)+1)

________________________________________________________________________________________

maxima [A]  time = 1.15, size = 31, normalized size = 0.82 \[ -\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {sech}\relax (x)\right ) + x \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ -\int \ln \left (\mathrm {cosh}\relax (x)\right )-\ln \relax (a) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a/cosh(x)),x)

[Out]

-int(log(cosh(x)) - log(a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*sech(x)),x)

[Out]

Integral(log(a*sech(x)), x)

________________________________________________________________________________________