3.92.1 \(\int \frac {2 x \log (x)+(-x+(x+x^2-x^3) \log (x)) \log (2 x^2)+(-2 \log (x)+(-1+x^2) \log (x) \log (2 x^2)) \log (\log (x))+(2 x^2 \log (x)-2 x \log (x) \log (\log (x))) \log (\frac {e^x}{-x+\log (\log (x))})}{-x^3 \log (5) \log (x) \log ^2(2 x^2)+x^2 \log (5) \log (x) \log ^2(2 x^2) \log (\log (x))} \, dx\)

Optimal. Leaf size=31 \[ \frac {\frac {1}{x}+\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{\log (5) \log \left (2 x^2\right )} \]

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Rubi [F]  time = 3.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x*Log[x] + (-x + (x + x^2 - x^3)*Log[x])*Log[2*x^2] + (-2*Log[x] + (-1 + x^2)*Log[x]*Log[2*x^2])*Log[Lo
g[x]] + (2*x^2*Log[x] - 2*x*Log[x]*Log[Log[x]])*Log[E^x/(-x + Log[Log[x]])])/(-(x^3*Log[5]*Log[x]*Log[2*x^2]^2
) + x^2*Log[5]*Log[x]*Log[2*x^2]^2*Log[Log[x]]),x]

[Out]

(x*ExpIntegralEi[Log[2*x^2]/2])/(2*Sqrt[2]*Sqrt[x^2]*Log[5]) + 1/(x*Log[5]*Log[2*x^2]) - Defer[Int][1/(Log[2*x
^2]*(x - Log[Log[x]])), x]/Log[5] + Defer[Int][1/(x*Log[x]*Log[2*x^2]*(x - Log[Log[x]])), x]/Log[5] - (2*Defer
[Int][Log[E^x/(-x + Log[Log[x]])]/(x*Log[2*x^2]^2), x])/Log[5]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x \log (x)-\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )-\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))-\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx\\ &=\frac {\int \frac {-2 x \log (x)-\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )-\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))-\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}\\ &=\frac {\int \left (\frac {-2 x \log (x)+x \log \left (2 x^2\right )-x \log (x) \log \left (2 x^2\right )-x^2 \log (x) \log \left (2 x^2\right )+x^3 \log (x) \log \left (2 x^2\right )+2 \log (x) \log (\log (x))+\log (x) \log \left (2 x^2\right ) \log (\log (x))-x^2 \log (x) \log \left (2 x^2\right ) \log (\log (x))}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))}-\frac {2 \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )}\right ) \, dx}{\log (5)}\\ &=\frac {\int \frac {-2 x \log (x)+x \log \left (2 x^2\right )-x \log (x) \log \left (2 x^2\right )-x^2 \log (x) \log \left (2 x^2\right )+x^3 \log (x) \log \left (2 x^2\right )+2 \log (x) \log (\log (x))+\log (x) \log \left (2 x^2\right ) \log (\log (x))-x^2 \log (x) \log \left (2 x^2\right ) \log (\log (x))}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \frac {x \log \left (2 x^2\right )+\log (x) \left (-2 x+2 \log (\log (x))+\log \left (2 x^2\right ) \left (x \left (-1-x+x^2\right )-\left (-1+x^2\right ) \log (\log (x))\right )\right )}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \left (\frac {-2-\log \left (2 x^2\right )+x^2 \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )}+\frac {1-x \log (x)}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))}\right ) \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \frac {-2-\log \left (2 x^2\right )+x^2 \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\int \frac {1-x \log (x)}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \frac {-2+\left (-1+x^2\right ) \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\int \left (-\frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))}+\frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))}\right ) \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \left (-\frac {2}{x^2 \log ^2\left (2 x^2\right )}+\frac {-1+x^2}{x^2 \log \left (2 x^2\right )}\right ) \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {\int \frac {-1+x^2}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {1}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {1}{x \log (5) \log \left (2 x^2\right )}+\frac {\int \left (\frac {1}{\log \left (2 x^2\right )}-\frac {1}{x^2 \log \left (2 x^2\right )}\right ) \, dx}{\log (5)}+\frac {\int \frac {1}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ &=\frac {1}{x \log (5) \log \left (2 x^2\right )}+\frac {\int \frac {1}{\log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)}\\ &=\frac {\sqrt {x^2} \text {Ei}\left (-\frac {1}{2} \log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)}+\frac {1}{x \log (5) \log \left (2 x^2\right )}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {x \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{2 \sqrt {2} \sqrt {x^2} \log (5)}-\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)}\\ &=\frac {x \text {Ei}\left (\frac {1}{2} \log \left (2 x^2\right )\right )}{2 \sqrt {2} \sqrt {x^2} \log (5)}+\frac {1}{x \log (5) \log \left (2 x^2\right )}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 34, normalized size = 1.10 \begin {gather*} \frac {1+x \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log (5) \log \left (2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x*Log[x] + (-x + (x + x^2 - x^3)*Log[x])*Log[2*x^2] + (-2*Log[x] + (-1 + x^2)*Log[x]*Log[2*x^2])*
Log[Log[x]] + (2*x^2*Log[x] - 2*x*Log[x]*Log[Log[x]])*Log[E^x/(-x + Log[Log[x]])])/(-(x^3*Log[5]*Log[x]*Log[2*
x^2]^2) + x^2*Log[5]*Log[x]*Log[2*x^2]^2*Log[Log[x]]),x]

[Out]

(1 + x*Log[E^x/(-x + Log[Log[x]])])/(x*Log[5]*Log[2*x^2])

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fricas [A]  time = 0.55, size = 35, normalized size = 1.13 \begin {gather*} \frac {x \log \left (-\frac {e^{x}}{x - \log \left (\log \relax (x)\right )}\right ) + 1}{x \log \relax (5) \log \relax (2) + 2 \, x \log \relax (5) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)*log(log(x))+2*x^2*log(x))*log(exp(x)/(log(log(x))-x))+((x^2-1)*log(x)*log(2*x^2)-2*log
(x))*log(log(x))+((-x^3+x^2+x)*log(x)-x)*log(2*x^2)+2*x*log(x))/(x^2*log(5)*log(x)*log(2*x^2)^2*log(log(x))-x^
3*log(5)*log(x)*log(2*x^2)^2),x, algorithm="fricas")

[Out]

(x*log(-e^x/(x - log(log(x)))) + 1)/(x*log(5)*log(2) + 2*x*log(5)*log(x))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)*log(log(x))+2*x^2*log(x))*log(exp(x)/(log(log(x))-x))+((x^2-1)*log(x)*log(2*x^2)-2*log
(x))*log(log(x))+((-x^3+x^2+x)*log(x)-x)*log(2*x^2)+2*x*log(x))/(x^2*log(5)*log(x)*log(2*x^2)^2*log(log(x))-x^
3*log(5)*log(x)*log(2*x^2)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nos
tep near 0S

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maple [C]  time = 0.40, size = 293, normalized size = 9.45

method	result	size
risch	\(\frac {2 i \ln \left ({\mathrm e}^{x}\right )}{\ln \relax (5) \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )\right )}+\frac {-2 i x \ln \left (x -\ln \left (\ln \relax (x )\right )\right )+2 \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \relax (x )\right )}\right )^{2}+\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{x -\ln \left (\ln \relax (x )\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \relax (x )\right )}\right )-\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \relax (x )\right )}\right )^{2}-\pi x \,\mathrm {csgn}\left (\frac {i}{x -\ln \left (\ln \relax (x )\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \relax (x )\right )}\right )^{2}-\pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \relax (x )\right )}\right )^{3}-2 \pi x +2 i}{\ln \relax (5) \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )\right ) x}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(x)*ln(ln(x))+2*x^2*ln(x))*ln(exp(x)/(ln(ln(x))-x))+((x^2-1)*ln(x)*ln(2*x^2)-2*ln(x))*ln(ln(x))+(
(-x^3+x^2+x)*ln(x)-x)*ln(2*x^2)+2*x*ln(x))/(x^2*ln(5)*ln(x)*ln(2*x^2)^2*ln(ln(x))-x^3*ln(5)*ln(x)*ln(2*x^2)^2)
,x,method=_RETURNVERBOSE)

[Out]

2*I/ln(5)/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*ln(x))*ln(ex
p(x))+(-2*I*x*ln(x-ln(ln(x)))+2*Pi*x*csgn(I*exp(x)/(x-ln(ln(x))))^2+Pi*x*csgn(I*exp(x))*csgn(I/(x-ln(ln(x))))*
csgn(I*exp(x)/(x-ln(ln(x))))-Pi*x*csgn(I*exp(x))*csgn(I*exp(x)/(x-ln(ln(x))))^2-Pi*x*csgn(I/(x-ln(ln(x))))*csg
n(I*exp(x)/(x-ln(ln(x))))^2-Pi*x*csgn(I*exp(x)/(x-ln(ln(x))))^3-2*Pi*x+2*I)/ln(5)/(Pi*csgn(I*x)^2*csgn(I*x^2)-
2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*ln(x))/x

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maxima [A]  time = 0.49, size = 33, normalized size = 1.06 \begin {gather*} \frac {x^{2} - x \log \left (-x + \log \left (\log \relax (x)\right )\right ) + 1}{x \log \relax (5) \log \relax (2) + 2 \, x \log \relax (5) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)*log(log(x))+2*x^2*log(x))*log(exp(x)/(log(log(x))-x))+((x^2-1)*log(x)*log(2*x^2)-2*log
(x))*log(log(x))+((-x^3+x^2+x)*log(x)-x)*log(2*x^2)+2*x*log(x))/(x^2*log(5)*log(x)*log(2*x^2)^2*log(log(x))-x^
3*log(5)*log(x)*log(2*x^2)^2),x, algorithm="maxima")

[Out]

(x^2 - x*log(-x + log(log(x))) + 1)/(x*log(5)*log(2) + 2*x*log(5)*log(x))

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mupad [B]  time = 8.33, size = 34, normalized size = 1.10 \begin {gather*} \frac {x\,\ln \left (-\frac {{\mathrm {e}}^x}{x-\ln \left (\ln \relax (x)\right )}\right )+1}{x\,\ln \relax (5)\,\ln \left (2\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x^2)*(x - log(x)*(x + x^2 - x^3)) + log(log(x))*(2*log(x) - log(2*x^2)*log(x)*(x^2 - 1)) - log(-exp
(x)/(x - log(log(x))))*(2*x^2*log(x) - 2*x*log(log(x))*log(x)) - 2*x*log(x))/(x^3*log(5)*log(2*x^2)^2*log(x) -
x^2*log(log(x))*log(5)*log(2*x^2)^2*log(x)),x)

[Out]

(x*log(-exp(x)/(x - log(log(x)))) + 1)/(x*log(5)*log(2*x^2))

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sympy [A]  time = 1.75, size = 46, normalized size = 1.48 \begin {gather*} \frac {1}{2 x \log {\relax (5 )} \log {\relax (x )} + x \log {\relax (2 )} \log {\relax (5 )}} + \frac {\log {\left (\frac {e^{x}}{- x + \log {\left (\log {\relax (x )} \right )}} \right )}}{2 \log {\relax (5 )} \log {\relax (x )} + \log {\relax (2 )} \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(x)*ln(ln(x))+2*x**2*ln(x))*ln(exp(x)/(ln(ln(x))-x))+((x**2-1)*ln(x)*ln(2*x**2)-2*ln(x))*ln
(ln(x))+((-x**3+x**2+x)*ln(x)-x)*ln(2*x**2)+2*x*ln(x))/(x**2*ln(5)*ln(x)*ln(2*x**2)**2*ln(ln(x))-x**3*ln(5)*ln
(x)*ln(2*x**2)**2),x)

[Out]

1/(2*x*log(5)*log(x) + x*log(2)*log(5)) + log(exp(x)/(-x + log(log(x))))/(2*log(5)*log(x) + log(2)*log(5))

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