3.39.2 \(\int \frac {24 x+16 x^3+e^x (12+8 x^2)+(6 x-40 x^3+e^x (3 x-20 x^3)+(-8 x^3-4 e^x x^3) \log (x)) \log (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}) \log (\log (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2})) \log (\log ^2(\log (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2})))}{(-12 x^3+80 x^5+e^{2 x} (-3 x+20 x^3)+e^x (-12 x^2+80 x^4)+(4 e^{2 x} x^3+16 e^x x^4+16 x^5) \log (x)) \log (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}) \log (\log (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}))} \, dx\)

Optimal. Leaf size=31 \[ \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (2+\frac {e^x}{x}\right ) x} \]

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Rubi [F]  time = 15.24, 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 {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24*x + 16*x^3 + E^x*(12 + 8*x^2) + (6*x - 40*x^3 + E^x*(3*x - 20*x^3) + (-8*x^3 - 4*E^x*x^3)*Log[x])*Log[
(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]*Log[Log[(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]]*Log[Log[Log[(-3 + 20*x^2
+ 4*x^2*Log[x])/(4*x^2)]]^2])/((-12*x^3 + 80*x^5 + E^(2*x)*(-3*x + 20*x^3) + E^x*(-12*x^2 + 80*x^4) + (4*E^(2*
x)*x^3 + 16*E^x*x^4 + 16*x^5)*Log[x])*Log[(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]*Log[Log[(-3 + 20*x^2 + 4*x^2*L
og[x])/(4*x^2)]]),x]

[Out]

12*Defer[Int][1/(x*(E^x + 2*x)*(-3 + 20*x^2 + 4*x^2*Log[x])*Log[5 - 3/(4*x^2) + Log[x]]*Log[Log[5 - 3/(4*x^2)
+ Log[x]]]), x] + 8*Defer[Int][x/((E^x + 2*x)*(-3 + 20*x^2 + 4*x^2*Log[x])*Log[5 - 3/(4*x^2) + Log[x]]*Log[Log
[5 - 3/(4*x^2) + Log[x]]]), x] - 2*Defer[Int][Log[Log[Log[5 - 3/(4*x^2) + Log[x]]]^2]/(E^x + 2*x)^2, x] + 2*De
fer[Int][(x*Log[Log[Log[5 - 3/(4*x^2) + Log[x]]]^2])/(E^x + 2*x)^2, x] - Defer[Int][Log[Log[Log[5 - 3/(4*x^2)
+ Log[x]]]^2]/(E^x + 2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-24 x-16 x^3-4 e^x \left (3+2 x^2\right )+\left (2+e^x\right ) x \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right )^2 \left (3-20 x^2-4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx\\ &=\int \left (\frac {2 (-1+x) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}-\frac {-12-8 x^2-3 x \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+20 x^3 \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+4 x^3 \log (x) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}\right ) \, dx\\ &=2 \int \frac {(-1+x) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx-\int \frac {-12-8 x^2-3 x \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+20 x^3 \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+4 x^3 \log (x) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx\\ &=2 \int \left (-\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}+\frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}\right ) \, dx-\int \frac {-\frac {4 \left (3+2 x^2\right )}{x \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}+\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \, dx\\ &=-\left (2 \int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx\right )+2 \int \frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx-\int \left (-\frac {12}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}-\frac {8 x}{\left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}+\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x}\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx\right )+2 \int \frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx+8 \int \frac {x}{\left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx+12 \int \frac {1}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx-\int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 26, normalized size = 0.84 \begin {gather*} \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x + 16*x^3 + E^x*(12 + 8*x^2) + (6*x - 40*x^3 + E^x*(3*x - 20*x^3) + (-8*x^3 - 4*E^x*x^3)*Log[x]
)*Log[(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]*Log[Log[(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]]*Log[Log[Log[(-3 + 2
0*x^2 + 4*x^2*Log[x])/(4*x^2)]]^2])/((-12*x^3 + 80*x^5 + E^(2*x)*(-3*x + 20*x^3) + E^x*(-12*x^2 + 80*x^4) + (4
*E^(2*x)*x^3 + 16*E^x*x^4 + 16*x^5)*Log[x])*Log[(-3 + 20*x^2 + 4*x^2*Log[x])/(4*x^2)]*Log[Log[(-3 + 20*x^2 + 4
*x^2*Log[x])/(4*x^2)]]),x]

[Out]

Log[Log[Log[5 - 3/(4*x^2) + Log[x]]]^2]/(E^x + 2*x)

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fricas [A]  time = 0.98, size = 33, normalized size = 1.06 \begin {gather*} \frac {\log \left (\log \left (\log \left (\frac {4 \, x^{2} \log \relax (x) + 20 \, x^{2} - 3}{4 \, x^{2}}\right )\right )^{2}\right )}{2 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)*x^3-8*x^3)*log(x)+(-20*x^3+3*x)*exp(x)-40*x^3+6*x)*log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)
*log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))*log(log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))^2)+(8*x^2+12)*exp(x)+
16*x^3+24*x)/((4*exp(x)^2*x^3+16*exp(x)*x^4+16*x^5)*log(x)+(20*x^3-3*x)*exp(x)^2+(80*x^4-12*x^2)*exp(x)+80*x^5
-12*x^3)/log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)/log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)),x, algorithm="fricas")

[Out]

log(log(log(1/4*(4*x^2*log(x) + 20*x^2 - 3)/x^2))^2)/(2*x + e^x)

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giac [A]  time = 5.01, size = 37, normalized size = 1.19 \begin {gather*} \frac {\log \left (\log \left (-2 \, \log \relax (2) + \log \left (4 \, x^{2} \log \relax (x) + 20 \, x^{2} - 3\right ) - 2 \, \log \relax (x)\right )^{2}\right )}{2 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)*x^3-8*x^3)*log(x)+(-20*x^3+3*x)*exp(x)-40*x^3+6*x)*log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)
*log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))*log(log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))^2)+(8*x^2+12)*exp(x)+
16*x^3+24*x)/((4*exp(x)^2*x^3+16*exp(x)*x^4+16*x^5)*log(x)+(20*x^3-3*x)*exp(x)^2+(80*x^4-12*x^2)*exp(x)+80*x^5
-12*x^3)/log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)/log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)),x, algorithm="giac")

[Out]

log(log(-2*log(2) + log(4*x^2*log(x) + 20*x^2 - 3) - 2*log(x))^2)/(2*x + e^x)

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maple [C]  time = 2.41, size = 820, normalized size = 26.45

method	result	size
risch	\(\frac {2 \ln \left (\ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )\right )}{{\mathrm e}^{x}+2 x}-\frac {i \pi \,\mathrm {csgn}\left (i \ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )^{2}\right ) \left (\mathrm {csgn}\left (i \ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )\right )^{2}-2 \,\mathrm {csgn}\left (i \ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )^{2}\right ) \mathrm {csgn}\left (i \ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )\right )+\mathrm {csgn}\left (i \ln \left (-2 \ln \relax (x )+\ln \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (-\frac {3}{4}+\left (5+\ln \relax (x )\right ) x^{2}\right )\right )\right )}{2}\right )^{2}\right )^{2}\right )}{2 \left ({\mathrm e}^{x}+2 x \right )}\)	\(820\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*exp(x)*x^3-8*x^3)*ln(x)+(-20*x^3+3*x)*exp(x)-40*x^3+6*x)*ln(1/4*(4*x^2*ln(x)+20*x^2-3)/x^2)*ln(ln(1/
4*(4*x^2*ln(x)+20*x^2-3)/x^2))*ln(ln(ln(1/4*(4*x^2*ln(x)+20*x^2-3)/x^2))^2)+(8*x^2+12)*exp(x)+16*x^3+24*x)/((4
*exp(x)^2*x^3+16*exp(x)*x^4+16*x^5)*ln(x)+(20*x^3-3*x)*exp(x)^2+(80*x^4-12*x^2)*exp(x)+80*x^5-12*x^3)/ln(1/4*(
4*x^2*ln(x)+20*x^2-3)/x^2)/ln(ln(1/4*(4*x^2*ln(x)+20*x^2-3)/x^2)),x,method=_RETURNVERBOSE)

[Out]

2/(exp(x)+2*x)*ln(ln(-2*ln(x)+ln(-3/4+(5+ln(x))*x^2)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*
csgn(I/x^2*(-3/4+(5+ln(x))*x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I/x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*
x^2))+csgn(I*(-3/4+(5+ln(x))*x^2)))))-1/2*I*Pi*csgn(I*ln(-2*ln(x)+ln(-3/4+(5+ln(x))*x^2)+1/2*I*Pi*csgn(I*x^2)*
(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I/x^2*(-3/4+(5+ln(x))*x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I
/x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I*(-3/4+(5+ln(x))*x^2))))^2)*(csgn(I*ln(-2*ln(x)+ln(-3/4+(5+ln(
x))*x^2)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I/x^2*(-3/4+(5+ln(x))*x^2))*(-csgn(I/x^
2*(-3/4+(5+ln(x))*x^2))+csgn(I/x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I*(-3/4+(5+ln(x))*x^2)))))^2-2*cs
gn(I*ln(-2*ln(x)+ln(-3/4+(5+ln(x))*x^2)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I/x^2*(-
3/4+(5+ln(x))*x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I/x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I*
(-3/4+(5+ln(x))*x^2))))^2)*csgn(I*ln(-2*ln(x)+ln(-3/4+(5+ln(x))*x^2)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I
*x))^2-1/2*I*Pi*csgn(I/x^2*(-3/4+(5+ln(x))*x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I/x^2))*(-csgn(I/x^2*
(-3/4+(5+ln(x))*x^2))+csgn(I*(-3/4+(5+ln(x))*x^2)))))+csgn(I*ln(-2*ln(x)+ln(-3/4+(5+ln(x))*x^2)+1/2*I*Pi*csgn(
I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I/x^2*(-3/4+(5+ln(x))*x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))
+csgn(I/x^2))*(-csgn(I/x^2*(-3/4+(5+ln(x))*x^2))+csgn(I*(-3/4+(5+ln(x))*x^2))))^2)^2)/(exp(x)+2*x)

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maxima [A]  time = 0.62, size = 36, normalized size = 1.16 \begin {gather*} \frac {2 \, \log \left (\log \left (-2 \, \log \relax (2) + \log \left (4 \, x^{2} \log \relax (x) + 20 \, x^{2} - 3\right ) - 2 \, \log \relax (x)\right )\right )}{2 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)*x^3-8*x^3)*log(x)+(-20*x^3+3*x)*exp(x)-40*x^3+6*x)*log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)
*log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))*log(log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2))^2)+(8*x^2+12)*exp(x)+
16*x^3+24*x)/((4*exp(x)^2*x^3+16*exp(x)*x^4+16*x^5)*log(x)+(20*x^3-3*x)*exp(x)^2+(80*x^4-12*x^2)*exp(x)+80*x^5
-12*x^3)/log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)/log(log(1/4*(4*x^2*log(x)+20*x^2-3)/x^2)),x, algorithm="maxima")

[Out]

2*log(log(-2*log(2) + log(4*x^2*log(x) + 20*x^2 - 3) - 2*log(x)))/(2*x + e^x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {24\,x+{\mathrm {e}}^x\,\left (8\,x^2+12\right )+16\,x^3+\ln \left (\frac {x^2\,\ln \relax (x)+5\,x^2-\frac {3}{4}}{x^2}\right )\,\ln \left ({\ln \left (\ln \left (\frac {x^2\,\ln \relax (x)+5\,x^2-\frac {3}{4}}{x^2}\right )\right )}^2\right )\,\ln \left (\ln \left (\frac {x^2\,\ln \relax (x)+5\,x^2-\frac {3}{4}}{x^2}\right )\right )\,\left (6\,x+{\mathrm {e}}^x\,\left (3\,x-20\,x^3\right )-40\,x^3-\ln \relax (x)\,\left (4\,x^3\,{\mathrm {e}}^x+8\,x^3\right )\right )}{\ln \left (\frac {x^2\,\ln \relax (x)+5\,x^2-\frac {3}{4}}{x^2}\right )\,\ln \left (\ln \left (\frac {x^2\,\ln \relax (x)+5\,x^2-\frac {3}{4}}{x^2}\right )\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (3\,x-20\,x^3\right )+{\mathrm {e}}^x\,\left (12\,x^2-80\,x^4\right )-\ln \relax (x)\,\left (16\,x^4\,{\mathrm {e}}^x+4\,x^3\,{\mathrm {e}}^{2\,x}+16\,x^5\right )+12\,x^3-80\,x^5\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x + exp(x)*(8*x^2 + 12) + 16*x^3 + log((x^2*log(x) + 5*x^2 - 3/4)/x^2)*log(log(log((x^2*log(x) + 5*x^
2 - 3/4)/x^2))^2)*log(log((x^2*log(x) + 5*x^2 - 3/4)/x^2))*(6*x + exp(x)*(3*x - 20*x^3) - 40*x^3 - log(x)*(4*x
^3*exp(x) + 8*x^3)))/(log((x^2*log(x) + 5*x^2 - 3/4)/x^2)*log(log((x^2*log(x) + 5*x^2 - 3/4)/x^2))*(exp(2*x)*(
3*x - 20*x^3) + exp(x)*(12*x^2 - 80*x^4) - log(x)*(16*x^4*exp(x) + 4*x^3*exp(2*x) + 16*x^5) + 12*x^3 - 80*x^5)
),x)

[Out]

-int((24*x + exp(x)*(8*x^2 + 12) + 16*x^3 + log((x^2*log(x) + 5*x^2 - 3/4)/x^2)*log(log(log((x^2*log(x) + 5*x^
2 - 3/4)/x^2))^2)*log(log((x^2*log(x) + 5*x^2 - 3/4)/x^2))*(6*x + exp(x)*(3*x - 20*x^3) - 40*x^3 - log(x)*(4*x
^3*exp(x) + 8*x^3)))/(log((x^2*log(x) + 5*x^2 - 3/4)/x^2)*log(log((x^2*log(x) + 5*x^2 - 3/4)/x^2))*(exp(2*x)*(
3*x - 20*x^3) + exp(x)*(12*x^2 - 80*x^4) - log(x)*(16*x^4*exp(x) + 4*x^3*exp(2*x) + 16*x^5) + 12*x^3 - 80*x^5)
), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)*x**3-8*x**3)*ln(x)+(-20*x**3+3*x)*exp(x)-40*x**3+6*x)*ln(1/4*(4*x**2*ln(x)+20*x**2-3)/x
**2)*ln(ln(1/4*(4*x**2*ln(x)+20*x**2-3)/x**2))*ln(ln(ln(1/4*(4*x**2*ln(x)+20*x**2-3)/x**2))**2)+(8*x**2+12)*ex
p(x)+16*x**3+24*x)/((4*exp(x)**2*x**3+16*exp(x)*x**4+16*x**5)*ln(x)+(20*x**3-3*x)*exp(x)**2+(80*x**4-12*x**2)*
exp(x)+80*x**5-12*x**3)/ln(1/4*(4*x**2*ln(x)+20*x**2-3)/x**2)/ln(ln(1/4*(4*x**2*ln(x)+20*x**2-3)/x**2)),x)

[Out]

Timed out

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