\(\int \log (a \text {sech}^n(x)) \, dx\) [217]

Optimal result

Integrand size = 7, antiderivative size = 43 \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=-\frac {n x^2}{2}+n x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^n(x)\right )+\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3799, 2221, 2317, 2438} \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=x \log \left (a \text {sech}^n(x)\right )+\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{2 x}\right )-\frac {n x^2}{2}+n x \log \left (e^{2 x}+1\right ) \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 2628

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

Rule 3799

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=\frac {n x^2}{2}+n x \log \left (1+e^{-2 x}\right )+x \log \left (a \text {sech}^n(x)\right )-\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{-2 x}\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \ln \left (a \operatorname {sech}\left (x \right )^{n}\right )d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.14 \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=-\frac {1}{2} \, n x^{2} + n x \log \left (\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right ) + n x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + n x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + n {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + n {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + x \log \left (a\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=\int \log {\left (a \operatorname {sech}^{n}{\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

Integral(log(a*sech(x)**n), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=-\frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} n + x \log \left (a \operatorname {sech}\left (x\right )^{n}\right ) \]

[In]

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

[Out]

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

Giac [F]

\[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=\int { \log \left (a \operatorname {sech}\left (x\right )^{n}\right ) \,d x } \]

[In]

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

[Out]

integrate(log(a*sech(x)^n), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (a \text {sech}^n(x)\right ) \, dx=\int \ln \left (a\,{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^n\right ) \,d x \]

[In]

int(log(a*(1/cosh(x))^n),x)

[Out]

int(log(a*(1/cosh(x))^n), x)