Integrand size = 6, antiderivative size = 52 \[ \int \log (\log (x) \sin (x)) \, dx=\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\operatorname {LogIntegral}(x)+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \] Output:
1/2*I*x^2-x*ln(1-exp(2*I*x))+x*ln(ln(x)*sin(x))-Li(x)+1/2*I*polylog(2,exp( 2*I*x))
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int \log (\log (x) \sin (x)) \, dx=-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\operatorname {LogIntegral}(x)+\frac {1}{2} i \left (x^2+\operatorname {PolyLog}\left (2,e^{2 i x}\right )\right ) \] Input:
Integrate[Log[Log[x]*Sin[x]],x]
Output:
-(x*Log[1 - E^((2*I)*x)]) + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (I/2)* (x^2 + PolyLog[2, E^((2*I)*x)])
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3029, 2009}
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.
\(\displaystyle \int \log (\log (x) \sin (x)) \, dx\) |
\(\Big \downarrow \) 3029 |
\(\displaystyle x \log (\log (x) \sin (x))-\int \left (x \cot (x)+\frac {1}{\log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\operatorname {LogIntegral}(x)+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))\) |
Input:
Int[Log[Log[x]*Sin[x]],x]
Output:
(I/2)*x^2 - x*Log[1 - E^((2*I)*x)] + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (I/2)*PolyLog[2, E^((2*I)*x)]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simpl ify[D[u, x]/u], x], x] /; ProductQ[u]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.33 (sec) , antiderivative size = 368, normalized size of antiderivative = 7.08
method | result | size |
risch | \(-x \ln \left ({\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}+\frac {i x^{2}}{2}-i \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}-\ln \left (2\right ) x -\frac {i \pi x}{2}-\frac {i \pi {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{3} x}{2}+i \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+\frac {i \pi \,\operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) x}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) x}{2}+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )-\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3} x}{2}+\ln \left (\ln \left (x \right )\right ) x +\operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right )\) | \(368\) |
Input:
int(ln(ln(x)*sin(x)),x,method=_RETURNVERBOSE)
Output:
-x*ln(exp(I*x))+1/2*I*Pi*csgn(I*ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*sin(x))^2 *x-1/2*I*Pi*csgn(ln(x)*sin(x))*csgn(I*ln(x)*sin(x))^2*x+1/2*I*Pi*csgn(I*ln (x)*sin(x))^2*x+1/2*I*x^2-I*dilog(exp(I*x))+1/2*I*Pi*csgn(I*ln(x))*csgn(I* ln(x)*(exp(2*I*x)-1))^2*x+1/2*I*Pi*csgn(ln(x)*sin(x))^3*x+1/2*I*Pi*csgn(I* (exp(2*I*x)-1))*csgn(I*ln(x)*(exp(2*I*x)-1))^2*x+1/2*I*Pi*csgn(I*exp(-I*x) )*csgn(ln(x)*sin(x))^2*x-ln(2)*x-1/2*I*Pi*x-1/2*I*Pi*csgn(I*ln(x)*(exp(2*I *x)-1))^3*x+I*dilog(exp(I*x)+1)+1/2*I*Pi*csgn(ln(x)*sin(x))*csgn(I*ln(x)*s in(x))*x-1/2*I*Pi*csgn(I*(exp(2*I*x)-1))*csgn(I*ln(x))*csgn(I*ln(x)*(exp(2 *I*x)-1))*x-I*ln(exp(I*x))*ln(exp(2*I*x)-1)+1/2*I*Pi*csgn(I*exp(-I*x))*csg n(I*ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*sin(x))*x+I*ln(exp(I*x))*ln(exp(I*x)+ 1)-1/2*I*Pi*csgn(I*ln(x)*sin(x))^3*x+ln(ln(x))*x+Ei(1,-ln(x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (37) = 74\).
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.10 \[ \int \log (\log (x) \sin (x)) \, dx=x \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac {1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \operatorname {log\_integral}\left (x\right ) \] Input:
integrate(log(log(x)*sin(x)),x, algorithm="fricas")
Output:
x*log(log(x)*sin(x)) - 1/2*x*log(cos(x) + I*sin(x) + 1) - 1/2*x*log(cos(x) - I*sin(x) + 1) - 1/2*x*log(-cos(x) + I*sin(x) + 1) - 1/2*x*log(-cos(x) - I*sin(x) + 1) + 1/2*I*dilog(cos(x) + I*sin(x)) - 1/2*I*dilog(cos(x) - I*s in(x)) - 1/2*I*dilog(-cos(x) + I*sin(x)) + 1/2*I*dilog(-cos(x) - I*sin(x)) - log_integral(x)
\[ \int \log (\log (x) \sin (x)) \, dx=\int \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \] Input:
integrate(ln(ln(x)*sin(x)),x)
Output:
Integral(log(log(x)*sin(x)), x)
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int \log (\log (x) \sin (x)) \, dx=\frac {1}{2} \, {\left (i \, \pi - 2 \, \log \left (2\right )\right )} x - \frac {1}{2} i \, x^{2} + x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right ) + i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \] Input:
integrate(log(log(x)*sin(x)),x, algorithm="maxima")
Output:
1/2*(I*pi - 2*log(2))*x - 1/2*I*x^2 + x*log(log(x)) - Ei(log(x)) + I*dilog (-e^(I*x)) + I*dilog(e^(I*x))
\[ \int \log (\log (x) \sin (x)) \, dx=\int { \log \left (\log \left (x\right ) \sin \left (x\right )\right ) \,d x } \] Input:
integrate(log(log(x)*sin(x)),x, algorithm="giac")
Output:
integrate(log(log(x)*sin(x)), x)
Timed out. \[ \int \log (\log (x) \sin (x)) \, dx=\int \ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \] Input:
int(log(log(x)*sin(x)),x)
Output:
int(log(log(x)*sin(x)), x)
\[ \int \log (\log (x) \sin (x)) \, dx=\int \mathrm {log}\left (\mathrm {log}\left (x \right ) \sin \left (x \right )\right )d x \] Input:
int(log(log(x)*sin(x)),x)
Output:
int(log(log(x)*sin(x)),x)