\(\int \log (a \text {sech}^2(x)) \, dx\) [216]

Optimal result

Integrand size = 7, antiderivative size = 35 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^2(x)\right )+\operatorname {PolyLog}\left (2,-e^{2 x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=x \left (x+2 \log \left (1+e^{-2 x}\right )+\log \left (a \text {sech}^2(x)\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 x}\right ) \] Input:

Integrate[Log[a*Sech[x]^2],x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3028, 27, 3042, 26, 4201, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \log \left (a \text {sech}^2(x)\right ) \, dx\)

\(\Big \downarrow \) 3028

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-\int -2 x \tanh (x)dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int x \tanh (x)dx+x \log \left (a \text {sech}^2(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )+2 \int -i x \tan (i x)dx\)

\(\Big \downarrow \) 26

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-2 i \int x \tan (i x)dx\)

\(\Big \downarrow \) 4201

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-2 i \left (2 i \int \frac {e^{2 x} x}{1+e^{2 x}}dx-\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-2 i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 x}\right )dx\right )-\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-2 i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{4} \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}\right )-\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle x \log \left (a \text {sech}^2(x)\right )-2 i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} x \log \left (e^{2 x}+1\right )\right )-\frac {i x^2}{2}\right )\)

Input:

Int[Log[a*Sech[x]^2],x]
 

Output:

x*Log[a*Sech[x]^2] - (2*I)*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*x)])/2 + 
 PolyLog[2, -E^(2*x)]/4))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3028

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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] + Simp[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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.84 (sec) , antiderivative size = 480, normalized size of antiderivative = 13.71

method	result	size
risch	\(-x^{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{3} x}{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} x +\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i a \right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}-i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x +\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i a \right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) x}{2}+2 \operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )+2 \operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3} x}{2}+2 x \ln \left ({\mathrm e}^{x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )+x \ln \left (a \right )+2 \ln \left (2\right ) x\)	\(480\)

Input:

int(ln(a*sech(x)^2),x,method=_RETURNVERBOSE)
 

Output:

-x^2-1/2*I*Pi*csgn(I*a/(1+exp(2*x))^2*exp(2*x))^3*x+I*Pi*csgn(I*exp(x))*cs 
gn(I*exp(2*x))^2*x+1/2*I*Pi*csgn(I*(1+exp(2*x))^2)^3*x+1/2*I*Pi*csgn(I*a)* 
csgn(I*a/(1+exp(2*x))^2*exp(2*x))^2*x-I*Pi*csgn(I*(1+exp(2*x)))*csgn(I*(1+ 
exp(2*x))^2)^2*x+1/2*I*Pi*csgn(I*(1+exp(2*x)))^2*csgn(I*(1+exp(2*x))^2)*x- 
1/2*I*Pi*csgn(I*exp(2*x))*csgn(I/(1+exp(2*x))^2)*csgn(I/(1+exp(2*x))^2*exp 
(2*x))*x-1/2*I*Pi*csgn(I/(1+exp(2*x))^2*exp(2*x))^3*x+1/2*I*Pi*csgn(I*exp( 
2*x))*csgn(I/(1+exp(2*x))^2*exp(2*x))^2*x+1/2*I*Pi*csgn(I/(1+exp(2*x))^2)* 
csgn(I/(1+exp(2*x))^2*exp(2*x))^2*x-1/2*I*Pi*csgn(I*a)*csgn(I/(1+exp(2*x)) 
^2*exp(2*x))*csgn(I*a/(1+exp(2*x))^2*exp(2*x))*x+2*dilog(1+I*exp(x))+2*dil 
og(1-I*exp(x))-1/2*I*Pi*csgn(I*exp(x))^2*csgn(I*exp(2*x))*x+1/2*I*Pi*csgn( 
I/(1+exp(2*x))^2*exp(2*x))*csgn(I*a/(1+exp(2*x))^2*exp(2*x))^2*x-1/2*I*Pi* 
csgn(I*exp(2*x))^3*x+2*x*ln(exp(x))-2*ln(exp(x))*ln(1+exp(2*x))+2*ln(exp(x 
))*ln(1+I*exp(x))+2*ln(exp(x))*ln(1-I*exp(x))+x*ln(a)+2*ln(2)*x
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.03 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^{2} + x \log \left (\frac {4 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right )}\right ) + 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \, {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + 2 \, {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \] Input:

integrate(log(a*sech(x)^2),x, algorithm="fricas")
 

Output:

-x^2 + x*log(4*(a*cosh(x) + a*sinh(x))/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + 
sinh(x)^3 + (3*cosh(x)^2 + 1)*sinh(x) + 3*cosh(x))) + 2*x*log(I*cosh(x) + 
I*sinh(x) + 1) + 2*x*log(-I*cosh(x) - I*sinh(x) + 1) + 2*dilog(I*cosh(x) + 
 I*sinh(x)) + 2*dilog(-I*cosh(x) - I*sinh(x))
 

Sympy [F]

\[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=\int \log {\left (a \operatorname {sech}^{2}{\left (x \right )} \right )}\, dx \] Input:

integrate(ln(a*sech(x)**2),x)
 

Output:

Integral(log(a*sech(x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^{2} + x \log \left (a \operatorname {sech}\left (x\right )^{2}\right ) + 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \] Input:

integrate(log(a*sech(x)^2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=\int { \log \left (a \operatorname {sech}\left (x\right )^{2}\right ) \,d x } \] Input:

integrate(log(a*sech(x)^2),x, algorithm="giac")
 

Output:

integrate(log(a*sech(x)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-\int 2\,\ln \left (\mathrm {cosh}\left (x\right )\right )-\ln \left (a\right ) \,d x \] Input:

int(log(a/cosh(x)^2),x)
 

Output:

-int(2*log(cosh(x)) - log(a), x)
 

Reduce [F]

\[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=\int \mathrm {log}\left (\mathrm {sech}\left (x \right )^{2} a \right )d x \] Input:

int(log(a*sech(x)^2),x)
 

Output:

int(log(sech(x)**2*a),x)