Integrand size = 5, antiderivative size = 39 \[ \int \log (a \cosh (x)) \, dx=\frac {x^2}{2}-x \log \left (1+e^{2 x}\right )+x \log (a \cosh (x))-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \] Output:
1/2*x^2-x*ln(1+exp(2*x))+x*ln(a*cosh(x))-1/2*polylog(2,-exp(2*x))
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \log (a \cosh (x)) \, dx=x \log (a \cosh (x))+\frac {1}{2} \left (-x \left (x+2 \log \left (1+e^{-2 x}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 x}\right )\right ) \] Input:
Integrate[Log[a*Cosh[x]],x]
Output:
x*Log[a*Cosh[x]] + (-(x*(x + 2*Log[1 + E^(-2*x)])) + PolyLog[2, -E^(-2*x)] )/2
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3028, 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.
\(\displaystyle \int \log (a \cosh (x)) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log (a \cosh (x))-\int x \tanh (x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \log (a \cosh (x))-\int -i x \tan (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \log (a \cosh (x))+i \int x \tan (i x)dx\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle x \log (a \cosh (x))+i \left (2 i \int \frac {e^{2 x} x}{1+e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle x \log (a \cosh (x))+i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 x}\right )dx\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log (a \cosh (x))+i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{4} \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log (a \cosh (x))+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} x \log \left (e^{2 x}+1\right )\right )-\frac {i x^2}{2}\right )\) |
Input:
Int[Log[a*Cosh[x]],x]
Output:
x*Log[a*Cosh[x]] + I*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*x)])/2 + PolyL og[2, -E^(2*x)]/4))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.69 (sec) , antiderivative size = 321, normalized size of antiderivative = 8.23
method | result | size |
risch | \(-x \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i a \right ) {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) x}{2}-\ln \left (2\right ) x +x \ln \left (a \right )+\frac {x^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i a \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(321\) |
Input:
int(ln(a*cosh(x)),x,method=_RETURNVERBOSE)
Output:
-x*ln(exp(x))-1/2*I*Pi*csgn(I*a*(1+exp(2*x))*exp(-x))^3*x+1/2*I*Pi*csgn(I* exp(-x))*csgn(I*exp(-x)*(1+exp(2*x)))^2*x-1/2*I*Pi*csgn(I*exp(-x)*(1+exp(2 *x)))^3*x+1/2*I*Pi*csgn(I*exp(-x)*(1+exp(2*x)))*csgn(I*a*(1+exp(2*x))*exp( -x))^2*x+1/2*I*Pi*csgn(I*a)*csgn(I*a*(1+exp(2*x))*exp(-x))^2*x-1/2*I*Pi*cs gn(I*exp(-x))*csgn(I*(1+exp(2*x)))*csgn(I*exp(-x)*(1+exp(2*x)))*x-ln(2)*x+ x*ln(a)+1/2*x^2-1/2*I*Pi*csgn(I*a)*csgn(I*exp(-x)*(1+exp(2*x)))*csgn(I*a*( 1+exp(2*x))*exp(-x))*x+1/2*I*Pi*csgn(I*(1+exp(2*x)))*csgn(I*exp(-x)*(1+exp (2*x)))^2*x+ln(exp(x))*ln(1+exp(2*x))-ln(exp(x))*ln(1+I*exp(x))-ln(exp(x)) *ln(1-I*exp(x))-dilog(1+I*exp(x))-dilog(1-I*exp(x))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.67 \[ \int \log (a \cosh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \] Input:
integrate(log(a*cosh(x)),x, algorithm="fricas")
Output:
1/2*x^2 + x*log(a*cosh(x)) - x*log(I*cosh(x) + I*sinh(x) + 1) - x*log(-I*c osh(x) - I*sinh(x) + 1) - dilog(I*cosh(x) + I*sinh(x)) - dilog(-I*cosh(x) - I*sinh(x))
\[ \int \log (a \cosh (x)) \, dx=\int \log {\left (a \cosh {\left (x \right )} \right )}\, dx \] Input:
integrate(ln(a*cosh(x)),x)
Output:
Integral(log(a*cosh(x)), x)
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \log (a \cosh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \] Input:
integrate(log(a*cosh(x)),x, algorithm="maxima")
Output:
1/2*x^2 + x*log(a*cosh(x)) - x*log(e^(2*x) + 1) - 1/2*dilog(-e^(2*x))
\[ \int \log (a \cosh (x)) \, dx=\int { \log \left (a \cosh \left (x\right )\right ) \,d x } \] Input:
integrate(log(a*cosh(x)),x, algorithm="giac")
Output:
integrate(log(a*cosh(x)), x)
Timed out. \[ \int \log (a \cosh (x)) \, dx=\int \ln \left (a\,\mathrm {cosh}\left (x\right )\right ) \,d x \] Input:
int(log(a*cosh(x)),x)
Output:
int(log(a*cosh(x)), x)
\[ \int \log (a \cosh (x)) \, dx=\int \mathrm {log}\left (\cosh \left (x \right ) a \right )d x \] Input:
int(log(a*cosh(x)),x)
Output:
int(log(cosh(x)*a),x)