\(\int x \log (\log (x) \sin (x)) \, dx\) [306]

Optimal result

Integrand size = 8, antiderivative size = 80 \[ \int x \log (\log (x) \sin (x)) \, dx=\frac {i x^3}{6}-\frac {1}{2} \operatorname {ExpIntegralEi}(2 \log (x))-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log (\log (x) \sin (x))+\frac {1}{2} i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 i x}\right ) \] Output:

1/6*I*x^3-1/2*Ei(2*ln(x))-1/2*x^2*ln(1-exp(2*I*x))+1/2*x^2*ln(ln(x)*sin(x) 
)+1/2*I*x*polylog(2,exp(2*I*x))-1/4*polylog(3,exp(2*I*x))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int x \log (\log (x) \sin (x)) \, dx=\frac {1}{48} \left (i \pi ^3-8 i x^3-24 \operatorname {ExpIntegralEi}(2 \log (x))-24 x^2 \log \left (1-e^{-2 i x}\right )+24 x^2 \log (\log (x) \sin (x))-24 i x \operatorname {PolyLog}\left (2,e^{-2 i x}\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i x}\right )\right ) \] Input:

Integrate[x*Log[Log[x]*Sin[x]],x]
 

Output:

(I*Pi^3 - (8*I)*x^3 - 24*ExpIntegralEi[2*Log[x]] - 24*x^2*Log[1 - E^((-2*I 
)*x)] + 24*x^2*Log[Log[x]*Sin[x]] - (24*I)*x*PolyLog[2, E^((-2*I)*x)] - 12 
*PolyLog[3, E^((-2*I)*x)])/48
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3035, 27, 7293, 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.

\(\displaystyle \int x \log (\log (x) \sin (x)) \, dx\)

\(\Big \downarrow \) 3035

\(\displaystyle \frac {1}{2} x^2 \log (\log (x) \sin (x))-\int \frac {x (x \cot (x) \log (x)+1)}{2 \log (x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 \log (\log (x) \sin (x))-\frac {1}{2} \int \frac {x (x \cot (x) \log (x)+1)}{\log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} x^2 \log (\log (x) \sin (x))-\frac {1}{2} \int \left (\cot (x) x^2+\frac {x}{\log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \log (\log (x) \sin (x))+\frac {1}{2} \left (-\operatorname {ExpIntegralEi}(2 \log (x))+i x \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i x}\right )+\frac {i x^3}{3}-x^2 \log \left (1-e^{2 i x}\right )\right )\)

Input:

Int[x*Log[Log[x]*Sin[x]],x]
 

Output:

(x^2*Log[Log[x]*Sin[x]])/2 + ((I/3)*x^3 - ExpIntegralEi[2*Log[x]] - x^2*Lo 
g[1 - E^((2*I)*x)] + I*x*PolyLog[2, E^((2*I)*x)] - PolyLog[3, E^((2*I)*x)] 
/2)/2
 

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3035

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Simp[Log[u]   w, x 
] - Int[SimplifyIntegrand[w*Simplify[D[u, x]/u], x], x] /; InverseFunctionF 
reeQ[w, x]] /; ProductQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (warning: unable to verify)

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

Time = 0.55 (sec) , antiderivative size = 398, normalized size of antiderivative = 4.98

method	result	size
risch	\(-\frac {x^{2} \ln \left ({\mathrm e}^{i x}\right )}{2}+\frac {\left (-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 )+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}+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 )+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2}+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 )+i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2}+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}-i \pi {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{3}+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}+i \pi \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3}-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}-i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3}-i \pi -2 \ln \left (2\right )\right ) x^{2}}{4}+\frac {x^{2} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}-\frac {x^{2} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+i x \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )-\operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )-\frac {x^{2} \ln \left ({\mathrm e}^{i x}+1\right )}{2}+i x \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )-\operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )+\frac {\ln \left (\ln \left (x \right )\right ) x^{2}}{2}+\frac {\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )}{2}+\frac {i x^{3}}{6}\)	\(398\)

Input:

int(x*ln(ln(x)*sin(x)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*x^2*ln(exp(I*x))+1/4*(-I*Pi*csgn(I*(exp(2*I*x)-1))*csgn(I*ln(x))*csgn 
(I*ln(x)*(exp(2*I*x)-1))+I*Pi*csgn(I*(exp(2*I*x)-1))*csgn(I*ln(x)*(exp(2*I 
*x)-1))^2+I*Pi*csgn(I*exp(-I*x))*csgn(I*ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*s 
in(x))+I*Pi*csgn(I*exp(-I*x))*csgn(ln(x)*sin(x))^2+I*Pi*csgn(ln(x)*sin(x)) 
*csgn(I*ln(x)*sin(x))+I*Pi*csgn(I*ln(x)*sin(x))^2+I*Pi*csgn(I*ln(x))*csgn( 
I*ln(x)*(exp(2*I*x)-1))^2-I*Pi*csgn(I*ln(x)*(exp(2*I*x)-1))^3+I*Pi*csgn(I* 
ln(x)*(exp(2*I*x)-1))*csgn(ln(x)*sin(x))^2+I*Pi*csgn(ln(x)*sin(x))^3-I*Pi* 
csgn(ln(x)*sin(x))*csgn(I*ln(x)*sin(x))^2-I*Pi*csgn(I*ln(x)*sin(x))^3-I*Pi 
-2*ln(2))*x^2+1/2*x^2*ln(exp(2*I*x)-1)-1/2*x^2*ln(1-exp(I*x))+I*x*polylog( 
2,exp(I*x))-polylog(3,exp(I*x))-1/2*x^2*ln(exp(I*x)+1)+I*x*polylog(2,-exp( 
I*x))-polylog(3,-exp(I*x))+1/2*ln(ln(x))*x^2+1/2*Ei(1,-2*ln(x))+1/6*I*x^3
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (54) = 108\).

Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.18 \[ \int x \log (\log (x) \sin (x)) \, dx=\frac {1}{2} \, x^{2} \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac {1}{4} \, x^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, x {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, x {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} \, \operatorname {log\_integral}\left (x^{2}\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} \, {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} \, {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \] Input:

integrate(x*log(log(x)*sin(x)),x, algorithm="fricas")
 

Output:

1/2*x^2*log(log(x)*sin(x)) - 1/4*x^2*log(cos(x) + I*sin(x) + 1) - 1/4*x^2* 
log(cos(x) - I*sin(x) + 1) - 1/4*x^2*log(-cos(x) + I*sin(x) + 1) - 1/4*x^2 
*log(-cos(x) - I*sin(x) + 1) + 1/2*I*x*dilog(cos(x) + I*sin(x)) - 1/2*I*x* 
dilog(cos(x) - I*sin(x)) - 1/2*I*x*dilog(-cos(x) + I*sin(x)) + 1/2*I*x*dil 
og(-cos(x) - I*sin(x)) - 1/2*log_integral(x^2) - 1/2*polylog(3, cos(x) + I 
*sin(x)) - 1/2*polylog(3, cos(x) - I*sin(x)) - 1/2*polylog(3, -cos(x) + I* 
sin(x)) - 1/2*polylog(3, -cos(x) - I*sin(x))
 

Sympy [F]

\[ \int x \log (\log (x) \sin (x)) \, dx=\int x \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \] Input:

integrate(x*ln(ln(x)*sin(x)),x)
 

Output:

Integral(x*log(log(x)*sin(x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int x \log (\log (x) \sin (x)) \, dx=-\frac {1}{4} \, {\left (-i \, \pi + 2 \, \log \left (2\right )\right )} x^{2} - \frac {1}{3} i \, x^{3} + \frac {1}{2} \, x^{2} \log \left (\log \left (x\right )\right ) + i \, x {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, x {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) - \frac {1}{2} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - {\rm Li}_{3}(-e^{\left (i \, x\right )}) - {\rm Li}_{3}(e^{\left (i \, x\right )}) \] Input:

integrate(x*log(log(x)*sin(x)),x, algorithm="maxima")
 

Output:

-1/4*(-I*pi + 2*log(2))*x^2 - 1/3*I*x^3 + 1/2*x^2*log(log(x)) + I*x*dilog( 
-e^(I*x)) + I*x*dilog(e^(I*x)) - 1/2*Ei(2*log(x)) - polylog(3, -e^(I*x)) - 
 polylog(3, e^(I*x))
 

Giac [F]

\[ \int x \log (\log (x) \sin (x)) \, dx=\int { x \log \left (\log \left (x\right ) \sin \left (x\right )\right ) \,d x } \] Input:

integrate(x*log(log(x)*sin(x)),x, algorithm="giac")
 

Output:

integrate(x*log(log(x)*sin(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \log (\log (x) \sin (x)) \, dx=\int x\,\ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \] Input:

int(x*log(log(x)*sin(x)),x)
 

Output:

int(x*log(log(x)*sin(x)), x)
 

Reduce [F]

\[ \int x \log (\log (x) \sin (x)) \, dx=\int \mathrm {log}\left (\mathrm {log}\left (x \right ) \sin \left (x \right )\right ) x d x \] Input:

int(x*log(log(x)*sin(x)),x)
 

Output:

int(log(log(x)*sin(x))*x,x)