Integrand size = 5, antiderivative size = 39 \[ \int \log (a \sinh (x)) \, dx=\frac {x^2}{2}-x \log \left (1-e^{2 x}\right )+x \log (a \sinh (x))-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \log (a \sinh (x)) \, dx=-\frac {x^2}{2}-x \log \left (1-e^{-2 x}\right )+x \log (a \sinh (x))+\frac {1}{2} \operatorname {PolyLog}\left (2,e^{-2 x}\right ) \]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {3028, 3042, 26, 4199, 25, 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.
| \(\displaystyle \int \log (a \sinh (x)) \, dx\) |
| \(\Big \downarrow \) 3028 |
| \(\displaystyle x \log (a \sinh (x))-\int x \coth (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle x \log (a \sinh (x))-\int -i x \tan \left (i x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle x \log (a \sinh (x))+i \int x \tan \left (i x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle x \log (a \sinh (x))+i \left (2 i \int -\frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \log (a \sinh (x))+i \left (-2 i \int \frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle x \log (a \sinh (x))+i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle x \log (a \sinh (x))+i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle x \log (a \sinh (x))+i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
x*Log[a*Sinh[x]] + I*((-1/2*I)*x^2 - (2*I)*(-1/2*(x*Log[1 - E^(2*x)]) - Po lyLog[2, E^(2*x)]/4))
3.3.1.3.1 Defintions of rubi rules used
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_.) + Pi*(k_.) + (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 ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.06 (sec) , antiderivative size = 295, normalized size of antiderivative = 7.56
| method | result | size |
| risch | \(-x \ln \left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} x}{2}-\frac {i \pi {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{3} x}{2}-x \ln \left (2\right )+\ln \left (a \right ) x +\frac {x^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (-1+{\mathrm e}^{2 x}\right )\right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\ln \left ({\mathrm e}^{x}\right ) \ln \left (-1+{\mathrm e}^{2 x}\right )-\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )+\operatorname {dilog}\left ({\mathrm e}^{x}\right )\) | \(295\) |
-x*ln(exp(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)))*csgn(I*a*(-1+exp(2*x))*exp(-x))*csgn (I*a)*x+1/2*I*Pi*csgn(I*(-1+exp(2*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*(-1+exp(2*x)) *exp(-x))^3*x-x*ln(2)+ln(a)*x+1/2*x^2-1/2*I*Pi*csgn(I*(-1+exp(2*x)))*csgn( I*exp(-x))*csgn(I*exp(-x)*(-1+exp(2*x)))*x+1/2*I*Pi*csgn(I*a*(-1+exp(2*x)) *exp(-x))^2*csgn(I*a)*x+ln(exp(x))*ln(-1+exp(2*x))-dilog(1+exp(x))-ln(exp( x))*ln(1+exp(x))+dilog(exp(x))
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.46 \[ \int \log (a \sinh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \sinh \left (x\right )\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) - {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
1/2*x^2 + x*log(a*sinh(x)) - x*log(cosh(x) + sinh(x) + 1) - x*log(-cosh(x) - sinh(x) + 1) - dilog(cosh(x) + sinh(x)) - dilog(-cosh(x) - sinh(x))
\[ \int \log (a \sinh (x)) \, dx=\int \log {\left (a \sinh {\left (x \right )} \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \log (a \sinh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \sinh \left (x\right )\right ) - x \log \left (e^{x} + 1\right ) - x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{x}\right ) - {\rm Li}_2\left (e^{x}\right ) \]
\[ \int \log (a \sinh (x)) \, dx=\int { \log \left (a \sinh \left (x\right )\right ) \,d x } \]
Timed out. \[ \int \log (a \sinh (x)) \, dx=\int \ln \left (a\,\mathrm {sinh}\left (x\right )\right ) \,d x \]