Integrand size = 140, antiderivative size = 24 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\left (-e^2+\log (x)+x (2-3 \log (4)-4 \log (16)+\log (x))\right )^2 \] Output:
((ln(x)-22*ln(2)+2)*x+ln(x)-exp(2))^2
Leaf count is larger than twice the leaf count of optimal. \(152\) vs. \(2(24)=48\).
Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.33 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=2 \left (-2 e^2 x+2 x^2+\frac {1}{4} x^2 \log (16)+\frac {5}{4} x^2 \log (256)+2 e^2 x \log (2048)-\frac {5}{2} x^2 \log (4194304)+\frac {1}{2} x^2 \log (4) \log (4194304)+\frac {5}{2} x^2 \log (16) \log (4194304)-e^2 \log (x)+2 x \log (x)-e^2 x \log (x)+2 x^2 \log (x)-\frac {1}{2} x^2 \log (16) \log (x)-\frac {5}{2} x^2 \log (256) \log (x)-2 x \log (2048) \log (x)+\frac {\log ^2(x)}{2}+x \log ^2(x)+\frac {1}{2} x^2 \log ^2(x)\right ) \] Input:
Integrate[(E^2*(-2 - 6*x) + 4*x + 12*x^2 + (-6*x + 6*E^2*x - 30*x^2)*Log[4 ] + 18*x^2*Log[4]^2 + (-8*x + 8*E^2*x - 40*x^2 + 48*x^2*Log[4])*Log[16] + 32*x^2*Log[16]^2 + (2 + 8*x - 2*E^2*x + 10*x^2 + (-6*x - 12*x^2)*Log[4] + (-8*x - 16*x^2)*Log[16])*Log[x] + (2*x + 2*x^2)*Log[x]^2)/x,x]
Output:
2*(-2*E^2*x + 2*x^2 + (x^2*Log[16])/4 + (5*x^2*Log[256])/4 + 2*E^2*x*Log[2 048] - (5*x^2*Log[4194304])/2 + (x^2*Log[4]*Log[4194304])/2 + (5*x^2*Log[1 6]*Log[4194304])/2 - E^2*Log[x] + 2*x*Log[x] - E^2*x*Log[x] + 2*x^2*Log[x] - (x^2*Log[16]*Log[x])/2 - (5*x^2*Log[256]*Log[x])/2 - 2*x*Log[2048]*Log[ x] + Log[x]^2/2 + x*Log[x]^2 + (x^2*Log[x]^2)/2)
Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(24)=48\).
Time = 0.46 (sec) , antiderivative size = 160, normalized size of antiderivative = 6.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6, 6, 2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {12 x^2+32 x^2 \log ^2(16)+18 x^2 \log ^2(4)+\left (2 x^2+2 x\right ) \log ^2(x)+\left (10 x^2+\left (-16 x^2-8 x\right ) \log (16)+\left (-12 x^2-6 x\right ) \log (4)-2 e^2 x+8 x+2\right ) \log (x)+\log (16) \left (-40 x^2+48 x^2 \log (4)+8 e^2 x-8 x\right )+\left (-30 x^2+6 e^2 x-6 x\right ) \log (4)+4 x+e^2 (-6 x-2)}{x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {32 x^2 \log ^2(16)+x^2 \left (12+18 \log ^2(4)\right )+\left (2 x^2+2 x\right ) \log ^2(x)+\left (10 x^2+\left (-16 x^2-8 x\right ) \log (16)+\left (-12 x^2-6 x\right ) \log (4)-2 e^2 x+8 x+2\right ) \log (x)+\log (16) \left (-40 x^2+48 x^2 \log (4)+8 e^2 x-8 x\right )+\left (-30 x^2+6 e^2 x-6 x\right ) \log (4)+4 x+e^2 (-6 x-2)}{x}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^2 \left (12+18 \log ^2(4)+32 \log ^2(16)\right )+\left (2 x^2+2 x\right ) \log ^2(x)+\left (10 x^2+\left (-16 x^2-8 x\right ) \log (16)+\left (-12 x^2-6 x\right ) \log (4)-2 e^2 x+8 x+2\right ) \log (x)+\log (16) \left (-40 x^2+48 x^2 \log (4)+8 e^2 x-8 x\right )+\left (-30 x^2+6 e^2 x-6 x\right ) \log (4)+4 x+e^2 (-6 x-2)}{x}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {2 \left (x^2 (5-6 \log (4)-8 \log (16))+x \left (4-e^2-\log (4194304)\right )+1\right ) \log (x)}{x}+2 (x+1) \log ^2(x)+\frac {2 \left (e^2-x (2-4 \log (16)-\log (64))\right ) (-(x (3-4 \log (16)-\log (64)))-1)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2}{2}+x^2 \log ^2(x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)-x^2 \log (x)+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+4 x+2 x \log ^2(x)+\log ^2(x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)-4 x \log (x)+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 x \left (4-e^2-\log (4194304)\right )-2 e^2 \log (x)\) |
Input:
Int[(E^2*(-2 - 6*x) + 4*x + 12*x^2 + (-6*x + 6*E^2*x - 30*x^2)*Log[4] + 18 *x^2*Log[4]^2 + (-8*x + 8*E^2*x - 40*x^2 + 48*x^2*Log[4])*Log[16] + 32*x^2 *Log[16]^2 + (2 + 8*x - 2*E^2*x + 10*x^2 + (-6*x - 12*x^2)*Log[4] + (-8*x - 16*x^2)*Log[16])*Log[x] + (2*x + 2*x^2)*Log[x]^2)/x,x]
Output:
4*x + x^2/2 - (x^2*(5 - 6*Log[4] - 8*Log[16]))/2 + x^2*(2 - 4*Log[16] - Lo g[64])*(3 - 4*Log[16] - Log[64]) - 2*x*(4 - E^2 - Log[4194304]) + 2*x*(2 - E^2*(3 - 4*Log[16] - Log[64]) - Log[4194304]) - 2*E^2*Log[x] - 4*x*Log[x] - x^2*Log[x] + x^2*(5 - 6*Log[4] - 8*Log[16])*Log[x] + 2*x*(4 - E^2 - Log [4194304])*Log[x] + Log[x]^2 + 2*x*Log[x]^2 + x^2*Log[x]^2
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(19)=38\).
Time = 2.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42
| method | result | size |
| norman | \(\ln \left (x \right )^{2}+x^{2} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )+\left (-4 \,{\mathrm e}^{2}+44 \,{\mathrm e}^{2} \ln \left (2\right )\right ) x +\left (4-88 \ln \left (2\right )+484 \ln \left (2\right )^{2}\right ) x^{2}+\left (4-44 \ln \left (2\right )\right ) x^{2} \ln \left (x \right )+\left (4-44 \ln \left (2\right )-2 \,{\mathrm e}^{2}\right ) x \ln \left (x \right )+2 x \ln \left (x \right )^{2}\) | \(82\) |
| risch | \(\left (x^{2}+2 x +1\right ) \ln \left (x \right )^{2}+\left (-44 x^{2} \ln \left (2\right )-44 x \ln \left (2\right )-2 \,{\mathrm e}^{2} x +4 x^{2}+4 x \right ) \ln \left (x \right )+484 x^{2} \ln \left (2\right )^{2}+44 \ln \left (2\right ) x \,{\mathrm e}^{2}-88 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{2} \ln \left (x \right )-4 \,{\mathrm e}^{2} x +4 x^{2}\) | \(83\) |
| parallelrisch | \(484 x^{2} \ln \left (2\right )^{2}-44 x^{2} \ln \left (2\right ) \ln \left (x \right )+x^{2} \ln \left (x \right )^{2}-44 x \ln \left (2\right ) \ln \left (x \right )+44 \ln \left (2\right ) x \,{\mathrm e}^{2}-88 x^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}-2 x \,{\mathrm e}^{2} \ln \left (x \right )+4 x^{2} \ln \left (x \right )+\ln \left (x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )+4 x \ln \left (x \right )-4 \,{\mathrm e}^{2} x +4 x^{2}\) | \(95\) |
| default | \(484 x^{2} \ln \left (2\right )^{2}-88 \ln \left (2\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+4 x^{2}-44 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+44 \ln \left (2\right ) x \,{\mathrm e}^{2}-110 x^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-2 \,{\mathrm e}^{2} \left (x \ln \left (x \right )-x \right )-44 x \ln \left (2\right )-6 \,{\mathrm e}^{2} x +\ln \left (x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )\) | \(118\) |
| parts | \(484 x^{2} \ln \left (2\right )^{2}-88 \ln \left (2\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+4 x^{2}-44 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+44 \ln \left (2\right ) x \,{\mathrm e}^{2}-110 x^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-2 \,{\mathrm e}^{2} \left (x \ln \left (x \right )-x \right )-44 x \ln \left (2\right )-6 \,{\mathrm e}^{2} x +\ln \left (x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )\) | \(118\) |
Input:
int(((2*x^2+2*x)*ln(x)^2+(4*(-16*x^2-8*x)*ln(2)+2*(-12*x^2-6*x)*ln(2)-2*ex p(2)*x+10*x^2+8*x+2)*ln(x)+584*x^2*ln(2)^2+4*(96*x^2*ln(2)+8*exp(2)*x-40*x ^2-8*x)*ln(2)+2*(6*exp(2)*x-30*x^2-6*x)*ln(2)+(-6*x-2)*exp(2)+12*x^2+4*x)/ x,x,method=_RETURNVERBOSE)
Output:
ln(x)^2+x^2*ln(x)^2-2*exp(2)*ln(x)+(-4*exp(2)+44*exp(2)*ln(2))*x+(4-88*ln( 2)+484*ln(2)^2)*x^2+(4-44*ln(2))*x^2*ln(x)+(4-44*ln(2)-2*exp(2))*x*ln(x)+2 *x*ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.21 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 4 \, x e^{2} - 44 \, {\left (2 \, x^{2} - x e^{2}\right )} \log \left (2\right ) + 2 \, {\left (2 \, x^{2} - {\left (x + 1\right )} e^{2} - 22 \, {\left (x^{2} + x\right )} \log \left (2\right ) + 2 \, x\right )} \log \left (x\right ) \] Input:
integrate(((2*x^2+2*x)*log(x)^2+(4*(-16*x^2-8*x)*log(2)+2*(-12*x^2-6*x)*lo g(2)-2*exp(2)*x+10*x^2+8*x+2)*log(x)+584*x^2*log(2)^2+4*(96*x^2*log(2)+8*e xp(2)*x-40*x^2-8*x)*log(2)+2*(6*exp(2)*x-30*x^2-6*x)*log(2)+(-6*x-2)*exp(2 )+12*x^2+4*x)/x,x, algorithm="fricas")
Output:
484*x^2*log(2)^2 + (x^2 + 2*x + 1)*log(x)^2 + 4*x^2 - 4*x*e^2 - 44*(2*x^2 - x*e^2)*log(2) + 2*(2*x^2 - (x + 1)*e^2 - 22*(x^2 + x)*log(2) + 2*x)*log( x)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.62 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=x^{2} \left (- 88 \log {\left (2 \right )} + 4 + 484 \log {\left (2 \right )}^{2}\right ) + x \left (- 4 e^{2} + 44 e^{2} \log {\left (2 \right )}\right ) + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}^{2} + \left (- 44 x^{2} \log {\left (2 \right )} + 4 x^{2} - 44 x \log {\left (2 \right )} - 2 x e^{2} + 4 x\right ) \log {\left (x \right )} - 2 e^{2} \log {\left (x \right )} \] Input:
integrate(((2*x**2+2*x)*ln(x)**2+(4*(-16*x**2-8*x)*ln(2)+2*(-12*x**2-6*x)* ln(2)-2*exp(2)*x+10*x**2+8*x+2)*ln(x)+584*x**2*ln(2)**2+4*(96*x**2*ln(2)+8 *exp(2)*x-40*x**2-8*x)*ln(2)+2*(6*exp(2)*x-30*x**2-6*x)*ln(2)+(-6*x-2)*exp (2)+12*x**2+4*x)/x,x)
Output:
x**2*(-88*log(2) + 4 + 484*log(2)**2) + x*(-4*exp(2) + 44*exp(2)*log(2)) + (x**2 + 2*x + 1)*log(x)**2 + (-44*x**2*log(2) + 4*x**2 - 44*x*log(2) - 2* x*exp(2) + 4*x)*log(x) - 2*exp(2)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (21) = 42\).
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.62 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} + \frac {1}{2} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - 110 \, x^{2} \log \left (2\right ) + 44 \, x e^{2} \log \left (2\right ) + 5 \, x^{2} \log \left (x\right ) + 2 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + \frac {7}{2} \, x^{2} - 2 \, {\left (x \log \left (x\right ) - x\right )} e^{2} - 6 \, x e^{2} - 22 \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (2\right ) - 44 \, {\left (x \log \left (x\right ) - x\right )} \log \left (2\right ) - 44 \, x \log \left (2\right ) + 8 \, x \log \left (x\right ) - 2 \, e^{2} \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x \] Input:
integrate(((2*x^2+2*x)*log(x)^2+(4*(-16*x^2-8*x)*log(2)+2*(-12*x^2-6*x)*lo g(2)-2*exp(2)*x+10*x^2+8*x+2)*log(x)+584*x^2*log(2)^2+4*(96*x^2*log(2)+8*e xp(2)*x-40*x^2-8*x)*log(2)+2*(6*exp(2)*x-30*x^2-6*x)*log(2)+(-6*x-2)*exp(2 )+12*x^2+4*x)/x,x, algorithm="maxima")
Output:
484*x^2*log(2)^2 + 1/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 - 110*x^2*log(2) + 44*x*e^2*log(2) + 5*x^2*log(x) + 2*(log(x)^2 - 2*log(x) + 2)*x + 7/2*x^2 - 2*(x*log(x) - x)*e^2 - 6*x*e^2 - 22*(2*x^2*log(x) - x^2)*log(2) - 44*(x*l og(x) - x)*log(2) - 44*x*log(2) + 8*x*log(x) - 2*e^2*log(x) + log(x)^2 - 4 *x
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} - 44 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 88 \, x^{2} \log \left (2\right ) + 44 \, x e^{2} \log \left (2\right ) + 4 \, x^{2} \log \left (x\right ) - 2 \, x e^{2} \log \left (x\right ) - 44 \, x \log \left (2\right ) \log \left (x\right ) + 2 \, x \log \left (x\right )^{2} + 4 \, x^{2} - 4 \, x e^{2} + 4 \, x \log \left (x\right ) - 2 \, e^{2} \log \left (x\right ) + \log \left (x\right )^{2} \] Input:
integrate(((2*x^2+2*x)*log(x)^2+(4*(-16*x^2-8*x)*log(2)+2*(-12*x^2-6*x)*lo g(2)-2*exp(2)*x+10*x^2+8*x+2)*log(x)+584*x^2*log(2)^2+4*(96*x^2*log(2)+8*e xp(2)*x-40*x^2-8*x)*log(2)+2*(6*exp(2)*x-30*x^2-6*x)*log(2)+(-6*x-2)*exp(2 )+12*x^2+4*x)/x,x, algorithm="giac")
Output:
484*x^2*log(2)^2 - 44*x^2*log(2)*log(x) + x^2*log(x)^2 - 88*x^2*log(2) + 4 4*x*e^2*log(2) + 4*x^2*log(x) - 2*x*e^2*log(x) - 44*x*log(2)*log(x) + 2*x* log(x)^2 + 4*x^2 - 4*x*e^2 + 4*x*log(x) - 2*e^2*log(x) + log(x)^2
Time = 3.64 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\left (2\,x+\ln \left (x\right )-22\,x\,\ln \left (2\right )+x\,\ln \left (x\right )\right )\,\left (2\,x-2\,{\mathrm {e}}^2+\ln \left (x\right )-22\,x\,\ln \left (2\right )+x\,\ln \left (x\right )\right ) \] Input:
int((4*x + 584*x^2*log(2)^2 + log(x)^2*(2*x + 2*x^2) - 4*log(2)*(8*x - 8*x *exp(2) - 96*x^2*log(2) + 40*x^2) + log(x)*(8*x - 2*log(2)*(6*x + 12*x^2) - 4*log(2)*(8*x + 16*x^2) - 2*x*exp(2) + 10*x^2 + 2) - 2*log(2)*(6*x - 6*x *exp(2) + 30*x^2) + 12*x^2 - exp(2)*(6*x + 2))/x,x)
Output:
(2*x + log(x) - 22*x*log(2) + x*log(x))*(2*x - 2*exp(2) + log(x) - 22*x*lo g(2) + x*log(x))
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.08 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\mathrm {log}\left (x \right )^{2} x^{2}+2 \mathrm {log}\left (x \right )^{2} x +\mathrm {log}\left (x \right )^{2}-44 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{2}-44 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x -2 \,\mathrm {log}\left (x \right ) e^{2} x -2 \,\mathrm {log}\left (x \right ) e^{2}+4 \,\mathrm {log}\left (x \right ) x^{2}+4 \,\mathrm {log}\left (x \right ) x +484 \mathrm {log}\left (2\right )^{2} x^{2}+44 \,\mathrm {log}\left (2\right ) e^{2} x -88 \,\mathrm {log}\left (2\right ) x^{2}-4 e^{2} x +4 x^{2} \] Input:
int(((2*x^2+2*x)*log(x)^2+(4*(-16*x^2-8*x)*log(2)+2*(-12*x^2-6*x)*log(2)-2 *exp(2)*x+10*x^2+8*x+2)*log(x)+584*x^2*log(2)^2+4*(96*x^2*log(2)+8*exp(2)* x-40*x^2-8*x)*log(2)+2*(6*exp(2)*x-30*x^2-6*x)*log(2)+(-6*x-2)*exp(2)+12*x ^2+4*x)/x,x)
Output:
log(x)**2*x**2 + 2*log(x)**2*x + log(x)**2 - 44*log(x)*log(2)*x**2 - 44*lo g(x)*log(2)*x - 2*log(x)*e**2*x - 2*log(x)*e**2 + 4*log(x)*x**2 + 4*log(x) *x + 484*log(2)**2*x**2 + 44*log(2)*e**2*x - 88*log(2)*x**2 - 4*e**2*x + 4 *x**2