\(\int x^2 \log (\log (x) \sin (x)) \, dx\) [305]

Optimal result

Integrand size = 10, antiderivative size = 98 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {i x^4}{12}-\frac {1}{3} \operatorname {ExpIntegralEi}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 i x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (4,e^{2 i x}\right ) \] Output:

1/12*I*x^4-1/3*Ei(3*ln(x))-1/3*x^3*ln(1-exp(2*I*x))+1/3*x^3*ln(ln(x)*sin(x 
))+1/2*I*x^2*polylog(2,exp(2*I*x))-1/2*x*polylog(3,exp(2*I*x))-1/4*I*polyl 
og(4,exp(2*I*x))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {1}{192} i \left (\pi ^4-16 x^4+64 i \operatorname {ExpIntegralEi}(3 \log (x))+64 i x^3 \log \left (1-e^{-2 i x}\right )-64 i x^3 \log (\log (x) \sin (x))-96 x^2 \operatorname {PolyLog}\left (2,e^{-2 i x}\right )+96 i x \operatorname {PolyLog}\left (3,e^{-2 i x}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i x}\right )\right ) \] Input:

Integrate[x^2*Log[Log[x]*Sin[x]],x]
 

Output:

(I/192)*(Pi^4 - 16*x^4 + (64*I)*ExpIntegralEi[3*Log[x]] + (64*I)*x^3*Log[1 
 - E^((-2*I)*x)] - (64*I)*x^3*Log[Log[x]*Sin[x]] - 96*x^2*PolyLog[2, E^((- 
2*I)*x)] + (96*I)*x*PolyLog[3, E^((-2*I)*x)] + 48*PolyLog[4, E^((-2*I)*x)] 
)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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.

\(\displaystyle \int x^2 \log (\log (x) \sin (x)) \, dx\)

\(\Big \downarrow \) 3035

\(\displaystyle \frac {1}{3} x^3 \log (\log (x) \sin (x))-\int \frac {x^2 (x \cot (x) \log (x)+1)}{3 \log (x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int \frac {x^2 (x \cot (x) \log (x)+1)}{\log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int \left (\cot (x) x^3+\frac {x^2}{\log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{3} \left (-\operatorname {ExpIntegralEi}(3 \log (x))+\frac {3}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {3}{2} x \operatorname {PolyLog}\left (3,e^{2 i x}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{2 i x}\right )+\frac {i x^4}{4}-x^3 \log \left (1-e^{2 i x}\right )\right )\)

Input:

Int[x^2*Log[Log[x]*Sin[x]],x]
 

Output:

(x^3*Log[Log[x]*Sin[x]])/3 + ((I/4)*x^4 - ExpIntegralEi[3*Log[x]] - x^3*Lo 
g[1 - E^((2*I)*x)] + ((3*I)/2)*x^2*PolyLog[2, E^((2*I)*x)] - (3*x*PolyLog[ 
3, E^((2*I)*x)])/2 - ((3*I)/4)*PolyLog[4, E^((2*I)*x)])/3
 

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 = 0.74 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.35

method	result	size
risch	\(-\frac {x^{3} \ln \left ({\mathrm e}^{i x}\right )}{3}+\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^{3}}{6}+\frac {x^{3} \ln \left ({\mathrm e}^{2 i x}-1\right )}{3}-\frac {x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{3}+i x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )-2 x \operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )-2 i \operatorname {polylog}\left (4, {\mathrm e}^{i x}\right )-\frac {x^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{3}+i x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )-2 x \operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )-2 i \operatorname {polylog}\left (4, -{\mathrm e}^{i x}\right )+\frac {x^{3} \ln \left (\ln \left (x \right )\right )}{3}+\frac {\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )}{3}+\frac {i x^{4}}{12}\)	\(426\)

Input:

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

Output:

-1/3*x^3*ln(exp(I*x))+1/6*(-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^3+1/3*x^3*ln(exp(2*I*x)-1)-1/3*x^3*ln(1-exp(I*x))+I*x^2*polylo 
g(2,exp(I*x))-2*x*polylog(3,exp(I*x))-2*I*polylog(4,exp(I*x))-1/3*x^3*ln(e 
xp(I*x)+1)+I*x^2*polylog(2,-exp(I*x))-2*x*polylog(3,-exp(I*x))-2*I*polylog 
(4,-exp(I*x))+1/3*x^3*ln(ln(x))+1/3*Ei(1,-3*ln(x))+1/12*I*x^4
 

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. 234 vs. \(2 (65) = 130\).

Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.39 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {1}{3} \, x^{3} \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac {1}{6} \, x^{3} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, x^{2} {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x^{2} {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x^{2} {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, x^{2} {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{3} \, \operatorname {log\_integral}\left (x^{3}\right ) - i \, {\rm polylog}\left (4, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, {\rm polylog}\left (4, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + i \, {\rm polylog}\left (4, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - i \, {\rm polylog}\left (4, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \] Input:

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

Output:

1/3*x^3*log(log(x)*sin(x)) - 1/6*x^3*log(cos(x) + I*sin(x) + 1) - 1/6*x^3* 
log(cos(x) - I*sin(x) + 1) - 1/6*x^3*log(-cos(x) + I*sin(x) + 1) - 1/6*x^3 
*log(-cos(x) - I*sin(x) + 1) + 1/2*I*x^2*dilog(cos(x) + I*sin(x)) - 1/2*I* 
x^2*dilog(cos(x) - I*sin(x)) - 1/2*I*x^2*dilog(-cos(x) + I*sin(x)) + 1/2*I 
*x^2*dilog(-cos(x) - I*sin(x)) - x*polylog(3, cos(x) + I*sin(x)) - x*polyl 
og(3, cos(x) - I*sin(x)) - x*polylog(3, -cos(x) + I*sin(x)) - x*polylog(3, 
 -cos(x) - I*sin(x)) - 1/3*log_integral(x^3) - I*polylog(4, cos(x) + I*sin 
(x)) + I*polylog(4, cos(x) - I*sin(x)) + I*polylog(4, -cos(x) + I*sin(x)) 
- I*polylog(4, -cos(x) - I*sin(x))
 

Sympy [F]

\[ \int x^2 \log (\log (x) \sin (x)) \, dx=\int x^{2} \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \] Input:

integrate(x**2*ln(ln(x)*sin(x)),x)
 

Output:

Integral(x**2*log(log(x)*sin(x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=-\frac {1}{6} \, {\left (-i \, \pi + 2 \, \log \left (2\right )\right )} x^{3} - \frac {1}{4} i \, x^{4} + \frac {1}{3} \, x^{3} \log \left (\log \left (x\right )\right ) + i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) - 2 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 2 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) - \frac {1}{3} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - 2 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) - 2 i \, {\rm Li}_{4}(e^{\left (i \, x\right )}) \] Input:

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

Output:

-1/6*(-I*pi + 2*log(2))*x^3 - 1/4*I*x^4 + 1/3*x^3*log(log(x)) + I*x^2*dilo 
g(-e^(I*x)) + I*x^2*dilog(e^(I*x)) - 2*x*polylog(3, -e^(I*x)) - 2*x*polylo 
g(3, e^(I*x)) - 1/3*Ei(3*log(x)) - 2*I*polylog(4, -e^(I*x)) - 2*I*polylog( 
4, e^(I*x))
 

Giac [F]

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

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

Output:

integrate(x^2*log(log(x)*sin(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\int x^2\,\ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \] Input:

int(x^2*log(log(x)*sin(x)),x)
 

Output:

int(x^2*log(log(x)*sin(x)), x)
 

Reduce [F]

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

int(x^2*log(log(x)*sin(x)),x)
 

Output:

int(log(log(x)*sin(x))*x**2,x)