3.45.48 \(\int \frac {(-2+(2-6 x^2) \log (x)-4 x^2 \log ^2(x)) \log (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)})}{x \log (x)} \, dx\)

Optimal. Leaf size=27 \[ \log ^2\left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right ) \]

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Rubi [F]  time = 0.70, 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 {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2 + (2 - 6*x^2)*Log[x] - 4*x^2*Log[x]^2)*Log[(4*E^(-x^2 - x^2*Log[x])*x)/(5*Log[x])])/(x*Log[x]),x]

[Out]

(3*x^2)/2 - (15*x^4)/8 - 3*ExpIntegralEi[2*Log[x]] - (3*x^4*Log[x])/2 - 3*x^2*Log[(4*x^(1 - x^2))/(5*E^x^2*Log
[x])] + 2*Defer[Int][Log[(4*x^(1 - x^2))/(5*E^x^2*Log[x])]/x, x] - 2*Defer[Int][Log[(4*x^(1 - x^2))/(5*E^x^2*L
og[x])]/(x*Log[x]), x] - 4*Defer[Int][x*Log[x]*Log[(4*x^(1 - x^2))/(5*E^x^2*Log[x])], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x}-6 x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )-\frac {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)}-4 x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx+6 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{2 \log (x)} \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \left (x-3 x^3-\frac {x}{\log (x)}-2 x^3 \log (x)\right ) \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=\frac {3 x^2}{2}-\frac {9 x^4}{4}-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \int \frac {x}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x^3 \log (x) \, dx\\ &=\frac {3 x^2}{2}-\frac {15 x^4}{8}-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=\frac {3 x^2}{2}-\frac {15 x^4}{8}-3 \text {Ei}(2 \log (x))-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 27, normalized size = 1.00 \begin {gather*} \log ^2\left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + (2 - 6*x^2)*Log[x] - 4*x^2*Log[x]^2)*Log[(4*E^(-x^2 - x^2*Log[x])*x)/(5*Log[x])])/(x*Log[x]),
x]

[Out]

Log[(4*x^(1 - x^2))/(5*E^x^2*Log[x])]^2

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fricas [A]  time = 0.57, size = 24, normalized size = 0.89 \begin {gather*} \log \left (\frac {4 \, x e^{\left (-x^{2} \log \relax (x) - x^{2}\right )}}{5 \, \log \relax (x)}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="f
ricas")

[Out]

log(4/5*x*e^(-x^2*log(x) - x^2)/log(x))^2

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giac [B]  time = 0.17, size = 86, normalized size = 3.19 \begin {gather*} x^{4} - 4 \, x^{2} \log \relax (2) + {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \relax (x)^{2} + 2 \, {\left (x^{4} - x^{2} {\left (2 \, \log \relax (2) + 1\right )}\right )} \log \relax (x) + 4 \, \log \relax (2) \log \relax (x) + 2 \, {\left (x^{2} \log \relax (x) + x^{2} - \log \relax (x)\right )} \log \left (5 \, \log \relax (x)\right ) + \log \left (5 \, \log \relax (x)\right )^{2} - 4 \, \log \relax (2) \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="g
iac")

[Out]

x^4 - 4*x^2*log(2) + (x^4 - 2*x^2 + 1)*log(x)^2 + 2*(x^4 - x^2*(2*log(2) + 1))*log(x) + 4*log(2)*log(x) + 2*(x
^2*log(x) + x^2 - log(x))*log(5*log(x)) + log(5*log(x))^2 - 4*log(2)*log(log(x))

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maple [C]  time = 0.35, size = 1507, normalized size = 55.81

method	result	size
risch	\(-4 \ln \relax (2) \ln \left (\ln \relax (x )\right )-4 x^{2} \ln \relax (2) \ln \relax (x )+2 \ln \left (\ln \relax (x )\right ) \ln \relax (5)+2 x^{2} \ln \relax (5)+\ln \left (\ln \relax (x )\right )^{2}+\ln \relax (x )^{2}-x^{4}+4 \ln \relax (2) \ln \relax (x )-x^{4} \ln \relax (x )^{2}-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )-2 x^{4} \ln \relax (x )-4 x^{2} \ln \relax (2)-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+2 x^{2} \ln \relax (5) \ln \relax (x )-2 \ln \relax (5) \ln \relax (x )+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+\left (2 x^{2} \ln \relax (x )+2 x^{2}-2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\right ) \ln \left (x^{x^{2}} {\mathrm e}^{x^{2}}\right )-i \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) x^{2} \pi +i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) x^{2} \pi -i \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) x^{2} \pi -i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )-i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )-i \pi \,x^{2} \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3} \ln \relax (x )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3} \ln \relax (x )-i \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{2} \pi +i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \ln \relax (x )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \ln \relax (x )\)	\(1507\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2*ln(x)^2+(-6*x^2+2)*ln(x)-2)*ln(4/5*x/ln(x)/exp(x^2*ln(x)+x^2))/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(2)*ln(ln(x))-4*x^2*ln(2)*ln(x)+2*ln(ln(x))*ln(5)+2*x^2*ln(5)+ln(ln(x))^2+ln(x)^2-x^4+4*ln(2)*ln(x)-x^4*l
n(x)^2-2*ln(x)*ln(ln(x))-2*x^4*ln(x)-4*x^2*ln(2)-I*Pi*ln(x)*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^3+I*Pi*ln(ln(x
))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^3-I*Pi*ln(x)*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^3+I*Pi*x^2*csgn(I*x/ln(x)/
(x^(x^2))*exp(-x^2))^3+I*Pi*x^2*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^3+I*Pi*ln(ln(x))*csgn(I*x/ln(x)/(x^(x^2))*ex
p(-x^2))^3+2*x^2*ln(5)*ln(x)-2*ln(5)*ln(x)+I*Pi*x^2*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^3*ln(x)+I*Pi*x^2*csgn(I*
x/ln(x)/(x^(x^2))*exp(-x^2))^3*ln(x)+(2*x^2*ln(x)+2*x^2-2*ln(x)+2*ln(ln(x)))*ln(x^(x^2)*exp(x^2))-I*Pi*ln(x)*c
sgn(I/ln(x))*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))-I*Pi*x^2*csgn(I*x)*csgn(I*x/ln(x)/(
x^(x^2))*exp(-x^2))^2*ln(x)-I*Pi*x^2*csgn(I/ln(x))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2*ln(x)+I*Pi*ln(ln(x))*cs
gn(I/ln(x))*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))+I*Pi*x^2*csgn(I*x)*csgn(I/ln(x)/(x^(
x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))+I*Pi*x^2*csgn(I/ln(x))*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I
/ln(x)/(x^(x^2))*exp(-x^2))-I*Pi*x^2*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2*ln(x)-I*P
i*x^2*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^2*ln(x)-I*Pi*ln(x)*csgn(I*x)*csgn(
I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))-I*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^2*csgn(
I*x)*x^2*Pi-I*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2*csgn(I/ln(x))*x^2*Pi+I*Pi*ln(x)*csgn(I*x)*csgn(I*x/ln(x)/(x^
(x^2))*exp(-x^2))^2+I*Pi*ln(x)*csgn(I/ln(x))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2-I*Pi*ln(ln(x))*csgn(I/ln(x))*
csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2-I*Pi*ln(ln(x))*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2
))^2-I*Pi*ln(ln(x))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^2+I*Pi*ln(ln(x))*csg
n(I*x)*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))-I*Pi*ln(ln(x))*csgn(I*x)*csgn(I*x
/ln(x)/(x^(x^2))*exp(-x^2))^2+I*Pi*ln(x)*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))^2+I*Pi*
ln(x)*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^2-I*csgn(I/ln(x)/(x^(x^2))*exp(-x^
2))^2*csgn(I/(x^(x^2))*exp(-x^2))*x^2*Pi-I*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))^2*csgn(I/ln(x)/(x^(x^2))*exp(-x
^2))*x^2*Pi+I*Pi*x^2*csgn(I*x)*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*csgn(I*x/ln(x)/(x^(x^2))*exp(-x^2))*ln(x)+I*P
i*x^2*csgn(I/ln(x))*csgn(I/(x^(x^2))*exp(-x^2))*csgn(I/ln(x)/(x^(x^2))*exp(-x^2))*ln(x)

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maxima [B]  time = 0.39, size = 100, normalized size = 3.70 \begin {gather*} -x^{4} - {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \relax (x)^{2} - 2 \, {\left (x^{4} - x^{2}\right )} \log \relax (x) - 2 \, {\left (x^{2} \log \relax (x) + x^{2} - \log \relax (x) + \log \left (\log \relax (x)\right )\right )} \log \left (\frac {4 \, x e^{\left (-x^{2} \log \relax (x) - x^{2}\right )}}{5 \, \log \relax (x)}\right ) - 2 \, {\left (x^{2} + {\left (x^{2} - 1\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="m
axima")

[Out]

-x^4 - (x^4 - 2*x^2 + 1)*log(x)^2 - 2*(x^4 - x^2)*log(x) - 2*(x^2*log(x) + x^2 - log(x) + log(log(x)))*log(4/5
*x*e^(-x^2*log(x) - x^2)/log(x)) - 2*(x^2 + (x^2 - 1)*log(x))*log(log(x)) - log(log(x))^2

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mupad [B]  time = 3.91, size = 23, normalized size = 0.85 \begin {gather*} {\ln \left (\frac {4\,x\,{\mathrm {e}}^{-x^2}}{5\,x^{x^2}\,\ln \relax (x)}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((4*x*exp(- x^2*log(x) - x^2))/(5*log(x)))*(4*x^2*log(x)^2 + log(x)*(6*x^2 - 2) + 2))/(x*log(x)),x)

[Out]

log((4*x*exp(-x^2))/(5*x^(x^2)*log(x)))^2

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sympy [A]  time = 0.57, size = 24, normalized size = 0.89 \begin {gather*} \log {\left (\frac {4 x e^{- x^{2} \log {\relax (x )} - x^{2}}}{5 \log {\relax (x )}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2*ln(x)**2+(-6*x**2+2)*ln(x)-2)*ln(4/5*x/ln(x)/exp(x**2*ln(x)+x**2))/x/ln(x),x)

[Out]

log(4*x*exp(-x**2*log(x) - x**2)/(5*log(x)))**2

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