\(\int \frac {8 x^2+8 \log (x)+(4 x^2+4 \log (x)) \log (x^2)+e^{\log ^2(x^2+\log (x))} (-4 x^2+(-4-4 x^2) \log (x)-4 \log ^2(x)+(-8-16 x^2) \log (x) \log (x^2+\log (x)))}{e^{2 \log ^2(x^2+\log (x))} (x^4 \log ^2(x)+x^2 \log ^3(x))+e^{\log ^2(x^2+\log (x))} (-2 x^4 \log (x)-2 x^2 \log ^2(x)) \log (x^2)+(x^4+x^2 \log (x)) \log ^2(x^2)} \, dx\) [3806]

Optimal result

Integrand size = 161, antiderivative size = 28 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \]

[Out]

4/x/(exp(ln(ln(x)+x^2)^2)*ln(x)-ln(x^2))

Rubi [F]

\[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(8*x^2 + 8*Log[x] + (4*x^2 + 4*Log[x])*Log[x^2] + E^Log[x^2 + Log[x]]^2*(-4*x^2 + (-4 - 4*x^2)*Log[x] - 4*
Log[x]^2 + (-8 - 16*x^2)*Log[x]*Log[x^2 + Log[x]]))/(E^(2*Log[x^2 + Log[x]]^2)*(x^4*Log[x]^2 + x^2*Log[x]^3) +
E^Log[x^2 + Log[x]]^2*(-2*x^4*Log[x] - 2*x^2*Log[x]^2)*Log[x^2] + (x^4 + x^2*Log[x])*Log[x^2]^2),x]

[Out]

8*Defer[Int][1/((x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2])^2), x] + 8*Defer[Int][Log[x]/(x^2*(x^
2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2])^2), x] - 4*Defer[Int][1/((x^2 + Log[x])*(E^Log[x^2 + Log
[x]]^2*Log[x] - Log[x^2])), x] - 4*Defer[Int][1/(x^2*(x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2]))
, x] - 4*Defer[Int][1/(Log[x]*(x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2])), x] - 4*Defer[Int][Log
[x]/(x^2*(x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2])), x] - 4*Defer[Int][Log[x^2]/(x^2*(x^2 + Log
[x])*(E^Log[x^2 + Log[x]]^2*Log[x] - Log[x^2])^2), x] - 4*Defer[Int][Log[x^2]/(Log[x]*(x^2 + Log[x])*(E^Log[x^
2 + Log[x]]^2*Log[x] - Log[x^2])^2), x] - 16*Defer[Int][Log[x^2 + Log[x]]/((x^2 + Log[x])*(E^Log[x^2 + Log[x]]
^2*Log[x] - Log[x^2])), x] - 8*Defer[Int][Log[x^2 + Log[x]]/(x^2*(x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x]
- Log[x^2])), x] - 16*Defer[Int][(Log[x^2]*Log[x^2 + Log[x]])/((x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x] -
Log[x^2])^2), x] - 8*Defer[Int][(Log[x^2]*Log[x^2 + Log[x]])/(x^2*(x^2 + Log[x])*(E^Log[x^2 + Log[x]]^2*Log[x]
- Log[x^2])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx \\ & = \int \left (-\frac {4 \left (x^2+\log (x)+x^2 \log (x)+\log ^2(x)+2 \log (x) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2+\log (x)\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}-\frac {4 \left (-2 x^2 \log (x)-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \log \left (x^2\right )+2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}\right ) \, dx \\ & = -\left (4 \int \frac {x^2+\log (x)+x^2 \log (x)+\log ^2(x)+2 \log (x) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2+\log (x)\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {-2 x^2 \log (x)-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \log \left (x^2\right )+2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx \\ & = -\left (4 \int \left (\frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {4 \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {2 \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}\right ) \, dx\right )-4 \int \frac {-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \left (-2 x^2+\log \left (x^2\right ) \left (1+\left (2+4 x^2\right ) \log \left (x^2+\log (x)\right )\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx \\ & = -\left (4 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \left (-\frac {2}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}-\frac {2 \log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {\log \left (x^2\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {\log \left (x^2\right )}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {4 \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {2 \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}\right ) \, dx-8 \int \frac {\log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-16 \int \frac {\log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx \\ & = -\left (4 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx+8 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx+8 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-8 \int \frac {\log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-8 \int \frac {\log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-16 \int \frac {\log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-16 \int \frac {\log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \]

[In]

Integrate[(8*x^2 + 8*Log[x] + (4*x^2 + 4*Log[x])*Log[x^2] + E^Log[x^2 + Log[x]]^2*(-4*x^2 + (-4 - 4*x^2)*Log[x
] - 4*Log[x]^2 + (-8 - 16*x^2)*Log[x]*Log[x^2 + Log[x]]))/(E^(2*Log[x^2 + Log[x]]^2)*(x^4*Log[x]^2 + x^2*Log[x
]^3) + E^Log[x^2 + Log[x]]^2*(-2*x^4*Log[x] - 2*x^2*Log[x]^2)*Log[x^2] + (x^4 + x^2*Log[x])*Log[x^2]^2),x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64

\[-\frac {8 i}{x \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \ln \left (x \right ) {\mathrm e}^{\ln \left (\ln \left (x \right )+x^{2}\right )^{2}}+4 i \ln \left (x \right )\right )}\]

[In]

int((((-16*x^2-8)*ln(x)*ln(ln(x)+x^2)-4*ln(x)^2+(-4*x^2-4)*ln(x)-4*x^2)*exp(ln(ln(x)+x^2)^2)+(4*ln(x)+4*x^2)*l
n(x^2)+8*ln(x)+8*x^2)/((x^2*ln(x)^3+x^4*ln(x)^2)*exp(ln(ln(x)+x^2)^2)^2+(-2*x^2*ln(x)^2-2*x^4*ln(x))*ln(x^2)*e
xp(ln(ln(x)+x^2)^2)+(x^2*ln(x)+x^4)*ln(x^2)^2),x)

[Out]

-8*I/x/(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(x)*exp(ln(ln(x)+x^2)^2
)+4*I*ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \]

[In]

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*l
og(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2
*x^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\log {\left (x^{2} + \log {\left (x \right )} \right )}^{2}} \log {\left (x \right )} - 2 x \log {\left (x \right )}} \]

[In]

integrate((((-16*x**2-8)*ln(x)*ln(ln(x)+x**2)-4*ln(x)**2+(-4*x**2-4)*ln(x)-4*x**2)*exp(ln(ln(x)+x**2)**2)+(4*l
n(x)+4*x**2)*ln(x**2)+8*ln(x)+8*x**2)/((x**2*ln(x)**3+x**4*ln(x)**2)*exp(ln(ln(x)+x**2)**2)**2+(-2*x**2*ln(x)*
*2-2*x**4*ln(x))*ln(x**2)*exp(ln(ln(x)+x**2)**2)+(x**2*ln(x)+x**4)*ln(x**2)**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \]

[In]

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*l
og(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2
*x^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.80 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \]

[In]

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*l
og(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2
*x^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 310, normalized size of antiderivative = 11.07 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4\,\left (2\,x^2\,\ln \left (x\right )+2\,{\ln \left (x\right )}^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2-4\,\ln \left (x^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2\,\left (\ln \left (x\right )+2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )+x^2+4\,x^2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\right )}{x^2\,\left (\ln \left (x\right )+x^2\right )\,\left (\ln \left (x^2\right )-{\mathrm {e}}^{{\ln \left (\ln \left (x\right )+x^2\right )}^2}\,\ln \left (x\right )\right )\,\left (x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+x\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+8\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+8\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+4\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+2\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )} \]

[In]

int((8*log(x) - exp(log(log(x) + x^2)^2)*(4*log(x)^2 + 4*x^2 + log(x)*(4*x^2 + 4) + log(log(x) + x^2)*log(x)*(
16*x^2 + 8)) + log(x^2)*(4*log(x) + 4*x^2) + 8*x^2)/(log(x^2)^2*(x^2*log(x) + x^4) + exp(2*log(log(x) + x^2)^2
)*(x^2*log(x)^3 + x^4*log(x)^2) - log(x^2)*exp(log(log(x) + x^2)^2)*(2*x^4*log(x) + 2*x^2*log(x)^2)),x)

[Out]

(4*(2*x^2*log(x) + 2*log(x)^2)*(x*log(x) + x^3)^2 - 4*log(x^2)*(x*log(x) + x^3)^2*(log(x) + 2*log(log(x) + x^2
)*log(x) + x^2 + 4*x^2*log(log(x) + x^2)*log(x)))/(x^2*(log(x) + x^2)*(log(x^2) - exp(log(log(x) + x^2)^2)*log
(x))*(x^5*(log(x^2) - 2*log(x)) + x*log(x)^2*(log(x^2) - 2*log(x)) + 2*x^3*log(x)*(log(x^2) - 2*log(x)) + 4*x^
3*log(log(x) + x^2)*log(x)^2 + 8*x^3*log(log(x) + x^2)*log(x)^3 + 8*x^5*log(log(x) + x^2)*log(x)^2 + 4*x*log(l
og(x) + x^2)*log(x)^3 + 2*x*log(log(x) + x^2)*log(x)^2*(log(x^2) - 2*log(x)) + 2*x^3*log(log(x) + x^2)*log(x)*
(log(x^2) - 2*log(x)) + 4*x^5*log(log(x) + x^2)*log(x)*(log(x^2) - 2*log(x)) + 4*x^3*log(log(x) + x^2)*log(x)^
2*(log(x^2) - 2*log(x))))