3.2.57 \(\int \frac {(-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} (15 x^2+7 x^3-2 x^4)+e^x (40 x^3+2 x^4-2 x^5)+(-5+x) \log (4)) \log (x)+(-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+(4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)) \log (x)) \log (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4))+(-4+2 x+5 x^4+e^{2 x} (3 x^2+2 x^3)+e^x (8 x^3+2 x^4)-\log (4)) \log (x) \log (\log (x))}{(-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)) \log (x)} \, dx\)

Optimal. Leaf size=30 \[ \log \left (x \left (-4+x+x^2 \left (e^x+x\right )^2-\log (4)\right )\right ) (5-x+\log (\log (x))) \]

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Rubi [F]  time = 13.62, 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 (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-20 + 14*x - 2*x^2 + 25*x^4 - 5*x^5 + E^(2*x)*(15*x^2 + 7*x^3 - 2*x^4) + E^x*(40*x^3 + 2*x^4 - 2*x^5) +
(-5 + x)*Log[4])*Log[x] + (-4 + x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - Log[4] + (4*x - x^2 - E^(2*x)*x^3 - 2*E^x*
x^4 - x^5 + x*Log[4])*Log[x])*Log[-4*x + x^2 + E^(2*x)*x^3 + 2*E^x*x^4 + x^5 - x*Log[4]] + (-4 + 2*x + 5*x^4 +
E^(2*x)*(3*x^2 + 2*x^3) + E^x*(8*x^3 + 2*x^4) - Log[4])*Log[x]*Log[Log[x]])/((-4*x + x^2 + E^(2*x)*x^3 + 2*E^
x*x^4 + x^5 - x*Log[4])*Log[x]),x]

[Out]

10*x + 12*Log[x] - x*Log[x^2 + E^(2*x)*x^3 + 2*E^x*x^4 + x^5 - 2*x*(2 + Log[2])] + 2*x*Log[Log[x]] + 3*Log[x]*
Log[Log[x]] - 2*LogIntegral[x] + 4*(2 + Log[2])*Defer[Int][(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/
2))^(-1), x] + (4*(2 + Log[2])*(27 + 4*Log[16])*Defer[Int][(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/
2))^(-1), x])/(8 + Log[16]) + 20*(2 + Log[2])*Defer[Int][1/(x*(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[
2]/2))), x] + (7 + Log[16])*Defer[Int][x/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 2*Defer[
Int][x^2/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 10*Defer[Int][(E^x*x^2)/(x + E^(2*x)*x^2
+ 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 10*Defer[Int][x^3/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log
[2]/2)), x] + 2*Defer[Int][(E^x*x^3)/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 2*Defer[Int]
[x^4/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 2*Defer[Int][(E^x*x^4)/(x + E^(2*x)*x^2 + 2*
E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 2*Defer[Int][x^5/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)
), x] + (17 + Log[16])*Defer[Int][x/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 2*Defer[Int]
[x^2/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 12*Defer[Int][(E^x*x^3)/(-x - E^(2*x)*x^2 -
2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 12*Defer[Int][x^4/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[
2]/2)), x] + 2*Defer[Int][(E^x*x^4)/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 2*Defer[Int]
[x^5/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + Defer[Int][Log[x^2 + E^(2*x)*x^3 + 2*E^x*x^
4 + x^5 - 4*x*(1 + Log[2]/2)]/(x*Log[x]), x] + (7 + Log[16])*Defer[Int][Log[Log[x]]/(x + E^(2*x)*x^2 + 2*E^x*x
^3 + x^4 - 4*(1 + Log[2]/2)), x] + 4*(2 + Log[2])*Defer[Int][Log[Log[x]]/(x*(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4
- 4*(1 + Log[2]/2))), x] + 2*Defer[Int][(E^x*x^2*Log[Log[x]])/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log
[2]/2)), x] + 2*Defer[Int][(x^3*Log[Log[x]])/(x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - 4*(1 + Log[2]/2)), x] + 2*De
fer[Int][(x*Log[Log[x]])/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 2*Defer[Int][(E^x*x^3*L
og[Log[x]])/(-x - E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x] + 2*Defer[Int][(x^4*Log[Log[x]])/(-x -
E^(2*x)*x^2 - 2*E^x*x^3 - x^4 + 4*(1 + Log[2]/2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (x^2+e^{2 x} x^3+2 e^x x^4+x^5+x (-4-\log (4))\right ) \log (x)} \, dx\\ &=\int \left (\frac {\left (2 x^2-2 e^x x^3-2 x^4+2 e^x x^4+2 x^5-8 \left (1+\frac {\log (2)}{2}\right )-7 x \left (1+\frac {4 \log (2)}{7}\right )\right ) (-5+x-\log (\log (x)))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )+3 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))}{x \log (x)}\right ) \, dx\\ &=\int \frac {\left (2 x^2-2 e^x x^3-2 x^4+2 e^x x^4+2 x^5-8 \left (1+\frac {\log (2)}{2}\right )-7 x \left (1+\frac {4 \log (2)}{7}\right )\right ) (-5+x-\log (\log (x)))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )} \, dx+\int \frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )+3 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))}{x \log (x)} \, dx\\ &=\int \left (\frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )}{x \log (x)}+\frac {(3+2 x) \log (\log (x))}{x}\right ) \, dx+\int \left (\frac {2 x^2}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {10 e^x x^2}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {10 x^3}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^4}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^5}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {12 e^x x^3}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {12 x^4}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {20 (2+\log (2))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {10 x \left (1+\frac {1}{10} (7+\log (16))\right )}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {4 (-2-\log (2)) \left (1-\frac {5 (7+\log (16))}{8+\log (16)}\right )}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^2 \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^3 \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^3 \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^4 \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {4 (2+\log (2)) \log (\log (x))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {(7+\log (16)) \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-20 + 14*x - 2*x^2 + 25*x^4 - 5*x^5 + E^(2*x)*(15*x^2 + 7*x^3 - 2*x^4) + E^x*(40*x^3 + 2*x^4 - 2*x
^5) + (-5 + x)*Log[4])*Log[x] + (-4 + x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - Log[4] + (4*x - x^2 - E^(2*x)*x^3 -
2*E^x*x^4 - x^5 + x*Log[4])*Log[x])*Log[-4*x + x^2 + E^(2*x)*x^3 + 2*E^x*x^4 + x^5 - x*Log[4]] + (-4 + 2*x + 5
*x^4 + E^(2*x)*(3*x^2 + 2*x^3) + E^x*(8*x^3 + 2*x^4) - Log[4])*Log[x]*Log[Log[x]])/((-4*x + x^2 + E^(2*x)*x^3
+ 2*E^x*x^4 + x^5 - x*Log[4])*Log[x]),x]

[Out]

Integrate[((-20 + 14*x - 2*x^2 + 25*x^4 - 5*x^5 + E^(2*x)*(15*x^2 + 7*x^3 - 2*x^4) + E^x*(40*x^3 + 2*x^4 - 2*x
^5) + (-5 + x)*Log[4])*Log[x] + (-4 + x + E^(2*x)*x^2 + 2*E^x*x^3 + x^4 - Log[4] + (4*x - x^2 - E^(2*x)*x^3 -
2*E^x*x^4 - x^5 + x*Log[4])*Log[x])*Log[-4*x + x^2 + E^(2*x)*x^3 + 2*E^x*x^4 + x^5 - x*Log[4]] + (-4 + 2*x + 5
*x^4 + E^(2*x)*(3*x^2 + 2*x^3) + E^x*(8*x^3 + 2*x^4) - Log[4])*Log[x]*Log[Log[x]])/((-4*x + x^2 + E^(2*x)*x^3
+ 2*E^x*x^4 + x^5 - x*Log[4])*Log[x]), x]

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fricas [B]  time = 1.11, size = 72, normalized size = 2.40 \begin {gather*} -{\left (x - 5\right )} \log \left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \relax (2) - 4 \, x\right ) + \log \left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \relax (2) - 4 \, x\right ) \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+3*x^2)*exp(x)^2+(2*x^4+8*x^3)*exp(x)-2*log(2)+5*x^4+2*x-4)*log(x)*log(log(x))+((-exp(x)^2*x
^3-2*exp(x)*x^4+2*x*log(2)-x^5-x^2+4*x)*log(x)+exp(x)^2*x^2+2*exp(x)*x^3-2*log(2)+x^4+x-4)*log(exp(x)^2*x^3+2*
exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)+((-2*x^4+7*x^3+15*x^2)*exp(x)^2+(-2*x^5+2*x^4+40*x^3)*exp(x)+2*(x-5)*log(2)
-5*x^5+25*x^4-2*x^2+14*x-20)*log(x))/(exp(x)^2*x^3+2*exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)/log(x),x, algorithm="f
ricas")

[Out]

-(x - 5)*log(x^5 + 2*x^4*e^x + x^3*e^(2*x) + x^2 - 2*x*log(2) - 4*x) + log(x^5 + 2*x^4*e^x + x^3*e^(2*x) + x^2
- 2*x*log(2) - 4*x)*log(log(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{4} + {\left (2 \, x^{3} + 3 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{x} + 2 \, x - 2 \, \log \relax (2) - 4\right )} \log \relax (x) \log \left (\log \relax (x)\right ) + {\left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} - {\left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \relax (2) - 4 \, x\right )} \log \relax (x) + x - 2 \, \log \relax (2) - 4\right )} \log \left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \relax (2) - 4 \, x\right ) - {\left (5 \, x^{5} - 25 \, x^{4} + 2 \, x^{2} + {\left (2 \, x^{4} - 7 \, x^{3} - 15 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} - x^{4} - 20 \, x^{3}\right )} e^{x} - 2 \, {\left (x - 5\right )} \log \relax (2) - 14 \, x + 20\right )} \log \relax (x)}{{\left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \relax (2) - 4 \, x\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+3*x^2)*exp(x)^2+(2*x^4+8*x^3)*exp(x)-2*log(2)+5*x^4+2*x-4)*log(x)*log(log(x))+((-exp(x)^2*x
^3-2*exp(x)*x^4+2*x*log(2)-x^5-x^2+4*x)*log(x)+exp(x)^2*x^2+2*exp(x)*x^3-2*log(2)+x^4+x-4)*log(exp(x)^2*x^3+2*
exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)+((-2*x^4+7*x^3+15*x^2)*exp(x)^2+(-2*x^5+2*x^4+40*x^3)*exp(x)+2*(x-5)*log(2)
-5*x^5+25*x^4-2*x^2+14*x-20)*log(x))/(exp(x)^2*x^3+2*exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)/log(x),x, algorithm="g
iac")

[Out]

integrate(((5*x^4 + (2*x^3 + 3*x^2)*e^(2*x) + 2*(x^4 + 4*x^3)*e^x + 2*x - 2*log(2) - 4)*log(x)*log(log(x)) + (
x^4 + 2*x^3*e^x + x^2*e^(2*x) - (x^5 + 2*x^4*e^x + x^3*e^(2*x) + x^2 - 2*x*log(2) - 4*x)*log(x) + x - 2*log(2)
- 4)*log(x^5 + 2*x^4*e^x + x^3*e^(2*x) + x^2 - 2*x*log(2) - 4*x) - (5*x^5 - 25*x^4 + 2*x^2 + (2*x^4 - 7*x^3 -
15*x^2)*e^(2*x) + 2*(x^5 - x^4 - 20*x^3)*e^x - 2*(x - 5)*log(2) - 14*x + 20)*log(x))/((x^5 + 2*x^4*e^x + x^3*
e^(2*x) + x^2 - 2*x*log(2) - 4*x)*log(x)), x)

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maple [C]  time = 0.56, size = 666, normalized size = 22.20

method	result	size
risch	\(\left (\ln \left (\ln \relax (x )\right )-x \right ) \ln \left (-\frac {x^{4}}{2}-{\mathrm e}^{x} x^{3}-\frac {{\mathrm e}^{2 x} x^{2}}{2}+\ln \relax (2)-\frac {x}{2}+2\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )-x \ln \relax (x )+\frac {i \pi x \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}}{2}-\frac {i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}}{2}-i \pi x -\frac {i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}}{2}+\frac {i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}}{2}-\frac {i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )}{2}+i \pi \ln \left (\ln \relax (x )\right )+15 \ln \relax (x )+5 \ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x -\frac {-x^{4}+2 \ln \relax (2)-x +4}{x^{2}}\right )-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )}{2}+i \pi x \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \relax (2)+\frac {x}{2}-2\right ) x \right )^{2}\)	\(666\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3+3*x^2)*exp(x)^2+(2*x^4+8*x^3)*exp(x)-2*ln(2)+5*x^4+2*x-4)*ln(x)*ln(ln(x))+((-exp(x)^2*x^3-2*exp(x
)*x^4+2*x*ln(2)-x^5-x^2+4*x)*ln(x)+exp(x)^2*x^2+2*exp(x)*x^3-2*ln(2)+x^4+x-4)*ln(exp(x)^2*x^3+2*exp(x)*x^4-2*x
*ln(2)+x^5+x^2-4*x)+((-2*x^4+7*x^3+15*x^2)*exp(x)^2+(-2*x^5+2*x^4+40*x^3)*exp(x)+2*(x-5)*ln(2)-5*x^5+25*x^4-2*
x^2+14*x-20)*ln(x))/(exp(x)^2*x^3+2*exp(x)*x^4-2*x*ln(2)+x^5+x^2-4*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.56, size = 81, normalized size = 2.70 \begin {gather*} -{\left (x - \log \left (\log \relax (x)\right )\right )} \log \left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} + x - 2 \, \log \relax (2) - 4\right ) - {\left (x - 15\right )} \log \relax (x) + \log \relax (x) \log \left (\log \relax (x)\right ) + 5 \, \log \left (\frac {x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} + x - 2 \, \log \relax (2) - 4}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+3*x^2)*exp(x)^2+(2*x^4+8*x^3)*exp(x)-2*log(2)+5*x^4+2*x-4)*log(x)*log(log(x))+((-exp(x)^2*x
^3-2*exp(x)*x^4+2*x*log(2)-x^5-x^2+4*x)*log(x)+exp(x)^2*x^2+2*exp(x)*x^3-2*log(2)+x^4+x-4)*log(exp(x)^2*x^3+2*
exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)+((-2*x^4+7*x^3+15*x^2)*exp(x)^2+(-2*x^5+2*x^4+40*x^3)*exp(x)+2*(x-5)*log(2)
-5*x^5+25*x^4-2*x^2+14*x-20)*log(x))/(exp(x)^2*x^3+2*exp(x)*x^4-2*x*log(2)+x^5+x^2-4*x)/log(x),x, algorithm="m
axima")

[Out]

-(x - log(log(x)))*log(x^4 + 2*x^3*e^x + x^2*e^(2*x) + x - 2*log(2) - 4) - (x - 15)*log(x) + log(x)*log(log(x)
) + 5*log((x^4 + 2*x^3*e^x + x^2*e^(2*x) + x - 2*log(2) - 4)/x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \relax (2)+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )\,\left (x-2\,\ln \relax (2)+2\,x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+x^4-\ln \relax (x)\,\left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \relax (2)+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )-4\right )+\ln \relax (x)\,\left (14\,x+{\mathrm {e}}^x\,\left (-2\,x^5+2\,x^4+40\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (-2\,x^4+7\,x^3+15\,x^2\right )+2\,\ln \relax (2)\,\left (x-5\right )-2\,x^2+25\,x^4-5\,x^5-20\right )+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (2\,x-2\,\ln \relax (2)+{\mathrm {e}}^x\,\left (2\,x^4+8\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (2\,x^3+3\,x^2\right )+5\,x^4-4\right )}{\ln \relax (x)\,\left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \relax (2)+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5)*(x - 2*log(2) + 2*x^3*exp(x) + x^2*exp(2*
x) + x^4 - log(x)*(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5) - 4) + log(x)*(14*x + exp(x)*(4
0*x^3 + 2*x^4 - 2*x^5) + exp(2*x)*(15*x^2 + 7*x^3 - 2*x^4) + 2*log(2)*(x - 5) - 2*x^2 + 25*x^4 - 5*x^5 - 20) +
log(log(x))*log(x)*(2*x - 2*log(2) + exp(x)*(8*x^3 + 2*x^4) + exp(2*x)*(3*x^2 + 2*x^3) + 5*x^4 - 4))/(log(x)*
(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5)),x)

[Out]

int((log(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5)*(x - 2*log(2) + 2*x^3*exp(x) + x^2*exp(2*
x) + x^4 - log(x)*(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5) - 4) + log(x)*(14*x + exp(x)*(4
0*x^3 + 2*x^4 - 2*x^5) + exp(2*x)*(15*x^2 + 7*x^3 - 2*x^4) + 2*log(2)*(x - 5) - 2*x^2 + 25*x^4 - 5*x^5 - 20) +
log(log(x))*log(x)*(2*x - 2*log(2) + exp(x)*(8*x^3 + 2*x^4) + exp(2*x)*(3*x^2 + 2*x^3) + 5*x^4 - 4))/(log(x)*
(2*x^4*exp(x) - 4*x - 2*x*log(2) + x^3*exp(2*x) + x^2 + x^5)), x)

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sympy [B]  time = 93.04, size = 76, normalized size = 2.53 \begin {gather*} \left (- x + \log {\left (\log {\relax (x )} \right )}\right ) \log {\left (x^{5} + 2 x^{4} e^{x} + x^{3} e^{2 x} + x^{2} - 4 x - 2 x \log {\relax (2 )} \right )} + 15 \log {\relax (x )} + 5 \log {\left (2 x e^{x} + e^{2 x} + \frac {x^{4} + x - 4 - 2 \log {\relax (2 )}}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3+3*x**2)*exp(x)**2+(2*x**4+8*x**3)*exp(x)-2*ln(2)+5*x**4+2*x-4)*ln(x)*ln(ln(x))+((-exp(x)**
2*x**3-2*exp(x)*x**4+2*x*ln(2)-x**5-x**2+4*x)*ln(x)+exp(x)**2*x**2+2*exp(x)*x**3-2*ln(2)+x**4+x-4)*ln(exp(x)**
2*x**3+2*exp(x)*x**4-2*x*ln(2)+x**5+x**2-4*x)+((-2*x**4+7*x**3+15*x**2)*exp(x)**2+(-2*x**5+2*x**4+40*x**3)*exp
(x)+2*(x-5)*ln(2)-5*x**5+25*x**4-2*x**2+14*x-20)*ln(x))/(exp(x)**2*x**3+2*exp(x)*x**4-2*x*ln(2)+x**5+x**2-4*x)
/ln(x),x)

[Out]

(-x + log(log(x)))*log(x**5 + 2*x**4*exp(x) + x**3*exp(2*x) + x**2 - 4*x - 2*x*log(2)) + 15*log(x) + 5*log(2*x
*exp(x) + exp(2*x) + (x**4 + x - 4 - 2*log(2))/x**2)

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