Integrand size = 116, antiderivative size = 24 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x+\left (3-\frac {e^{2+x^2}}{-4+e^3}+\log (x)\right )^2 \] Output:
(ln(x)+3-exp(x^2+2)/(exp(3)-4))^2+x
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(24)=48\).
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{4+2 x^2}-6 e^{2+x^2} \left (-4+e^3\right )+\left (-4+e^3\right )^2 x-2 e^{2+x^2} \left (-4+e^3\right ) \log (x)+6 \left (-4+e^3\right )^2 \log (x)+\left (-4+e^3\right )^2 \log ^2(x)}{\left (-4+e^3\right )^2} \] Input:
Integrate[(96 + E^3*(-48 - 8*x) + 16*x + 4*E^(4 + 2*x^2)*x^2 + E^6*(6 + x) + E^(2 + x^2)*(8 + 48*x^2 + E^3*(-2 - 12*x^2)) + (32 - 16*E^3 + 2*E^6 + E ^(2 + x^2)*(16*x^2 - 4*E^3*x^2))*Log[x])/(16*x - 8*E^3*x + E^6*x),x]
Output:
(E^(4 + 2*x^2) - 6*E^(2 + x^2)*(-4 + E^3) + (-4 + E^3)^2*x - 2*E^(2 + x^2) *(-4 + E^3)*Log[x] + 6*(-4 + E^3)^2*Log[x] + (-4 + E^3)^2*Log[x]^2)/(-4 + E^3)^2
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(24)=48\).
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.71, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {6, 6, 27, 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.
| \(\displaystyle \int \frac {4 e^{2 x^2+4} x^2+e^{x^2+2} \left (48 x^2+e^3 \left (-12 x^2-2\right )+8\right )+\left (e^{x^2+2} \left (16 x^2-4 e^3 x^2\right )+2 e^6-16 e^3+32\right ) \log (x)+16 x+e^3 (-8 x-48)+e^6 (x+6)+96}{e^6 x-8 e^3 x+16 x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 e^{2 x^2+4} x^2+e^{x^2+2} \left (48 x^2+e^3 \left (-12 x^2-2\right )+8\right )+\left (e^{x^2+2} \left (16 x^2-4 e^3 x^2\right )+2 e^6-16 e^3+32\right ) \log (x)+16 x+e^3 (-8 x-48)+e^6 (x+6)+96}{\left (16-8 e^3\right ) x+e^6 x}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 e^{2 x^2+4} x^2+e^{x^2+2} \left (48 x^2+e^3 \left (-12 x^2-2\right )+8\right )+\left (e^{x^2+2} \left (16 x^2-4 e^3 x^2\right )+2 e^6-16 e^3+32\right ) \log (x)+16 x+e^3 (-8 x-48)+e^6 (x+6)+96}{\left (16-8 e^3+e^6\right ) x}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 e^{2 x^2+4} x^2+16 x+e^6 (x+6)-8 e^3 (x+6)+2 e^{x^2+2} \left (24 x^2-e^3 \left (6 x^2+1\right )+4\right )+2 \left (2 e^{x^2+2} \left (4 x^2-e^3 x^2\right )+\left (-4+e^3\right )^2\right ) \log (x)+96}{x}dx}{\left (4-e^3\right )^2}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \frac {\int \frac {4 e^{2 x^2+4} x^2+16 x+\left (-8 e^3+e^6\right ) (x+6)+2 e^{x^2+2} \left (24 x^2-e^3 \left (6 x^2+1\right )+4\right )+2 \left (2 e^{x^2+2} \left (4 x^2-e^3 x^2\right )+\left (-4+e^3\right )^2\right ) \log (x)+96}{x}dx}{\left (4-e^3\right )^2}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\int \left (4 e^{2 x^2+4} x+\frac {\left (-4+e^3\right )^2 (x+2 \log (x)+6)}{x}-\frac {2 e^{x^2+2} \left (-4+e^3\right ) \left (2 \log (x) x^2+6 x^2+1\right )}{x}\right )dx}{\left (4-e^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^{2 x^2+4}+\frac {2 \left (4-e^3\right ) e^{x^2+2} \left (3 x^2+x^2 \log (x)\right )}{x^2}+\left (4-e^3\right )^2 x+\left (4-e^3\right )^2 \log ^2(x)+6 \left (4-e^3\right )^2 \log (x)}{\left (4-e^3\right )^2}\) |
Input:
Int[(96 + E^3*(-48 - 8*x) + 16*x + 4*E^(4 + 2*x^2)*x^2 + E^6*(6 + x) + E^( 2 + x^2)*(8 + 48*x^2 + E^3*(-2 - 12*x^2)) + (32 - 16*E^3 + 2*E^6 + E^(2 + x^2)*(16*x^2 - 4*E^3*x^2))*Log[x])/(16*x - 8*E^3*x + E^6*x),x]
Output:
(E^(4 + 2*x^2) + (4 - E^3)^2*x + 6*(4 - E^3)^2*Log[x] + (4 - E^3)^2*Log[x] ^2 + (2*E^(2 + x^2)*(4 - E^3)*(3*x^2 + x^2*Log[x]))/x^2)/(4 - E^3)^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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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. \(61\) vs. \(2(22)=44\).
Time = 3.61 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(x +6 \ln \left (x \right )-\frac {6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}+\ln \left (x \right )^{2}+\frac {{\mathrm e}^{2 x^{2}+4}}{{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+16}\) | \(62\) |
| parts | \(x +6 \ln \left (x \right )-\frac {6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}+\ln \left (x \right )^{2}+\frac {{\mathrm e}^{2 x^{2}+4}}{{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+16}\) | \(62\) |
| norman | \(\frac {\frac {{\mathrm e}^{2 x^{2}+4}}{{\mathrm e}^{3}-4}+\left ({\mathrm e}^{3}-4\right ) x +\left ({\mathrm e}^{3}-4\right ) \ln \left (x \right )^{2}+\left (6 \,{\mathrm e}^{3}-24\right ) \ln \left (x \right )-2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )-6 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{3}-4}\) | \(66\) |
| parallelrisch | \(\frac {\ln \left (x \right )^{2} {\mathrm e}^{6}+x \,{\mathrm e}^{6}+6 \,{\mathrm e}^{6} \ln \left (x \right )-8 \ln \left (x \right )^{2} {\mathrm e}^{3}-2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right ) {\mathrm e}^{3}-8 x \,{\mathrm e}^{3}-48 \ln \left (x \right ) {\mathrm e}^{3}-6 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}+2}+16 \ln \left (x \right )^{2}+8 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )+{\mathrm e}^{2 x^{2}+4}+16 x +96 \ln \left (x \right )+24 \,{\mathrm e}^{x^{2}+2}}{{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+16}\) | \(118\) |
| risch | \(\ln \left (x \right )^{2}-\frac {2 \,{\mathrm e}^{x^{2}+2} \ln \left (x \right )}{{\mathrm e}^{3}-4}-\frac {48 \ln \left (x \right ) {\mathrm e}^{3}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {6 \,{\mathrm e}^{6} \ln \left (x \right )}{\left ({\mathrm e}^{3}-4\right )^{2}}-\frac {8 \,{\mathrm e}^{3} x}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {x \,{\mathrm e}^{6}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {96 \ln \left (x \right )}{\left ({\mathrm e}^{3}-4\right )^{2}}-\frac {6 \,{\mathrm e}^{x^{2}+5}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {{\mathrm e}^{2 x^{2}+4}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {24 \,{\mathrm e}^{x^{2}+2}}{\left ({\mathrm e}^{3}-4\right )^{2}}+\frac {16 x}{\left ({\mathrm e}^{3}-4\right )^{2}}\) | \(129\) |
Input:
int((((-4*x^2*exp(3)+16*x^2)*exp(x^2+2)+2*exp(3)^2-16*exp(3)+32)*ln(x)+4*x ^2*exp(x^2+2)^2+((-12*x^2-2)*exp(3)+48*x^2+8)*exp(x^2+2)+(6+x)*exp(3)^2+(- 8*x-48)*exp(3)+16*x+96)/(x*exp(3)^2-8*x*exp(3)+16*x),x,method=_RETURNVERBO SE)
Output:
x+6*ln(x)-6*exp(x^2+2)/(exp(3)-4)-2/(exp(3)-4)*exp(x^2+2)*ln(x)+ln(x)^2+1/ (exp(3)^2-8*exp(3)+16)*exp(x^2+2)^2
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {{\left (e^{6} - 8 \, e^{3} + 16\right )} \log \left (x\right )^{2} + x e^{6} - 8 \, x e^{3} - 6 \, {\left (e^{3} - 4\right )} e^{\left (x^{2} + 2\right )} - 2 \, {\left ({\left (e^{3} - 4\right )} e^{\left (x^{2} + 2\right )} - 3 \, e^{6} + 24 \, e^{3} - 48\right )} \log \left (x\right ) + 16 \, x + e^{\left (2 \, x^{2} + 4\right )}}{e^{6} - 8 \, e^{3} + 16} \] Input:
integrate((((-4*x^2*exp(3)+16*x^2)*exp(x^2+2)+2*exp(3)^2-16*exp(3)+32)*log (x)+4*x^2*exp(x^2+2)^2+((-12*x^2-2)*exp(3)+48*x^2+8)*exp(x^2+2)+(6+x)*exp( 3)^2+(-8*x-48)*exp(3)+16*x+96)/(x*exp(3)^2-8*x*exp(3)+16*x),x, algorithm=" fricas")
Output:
((e^6 - 8*e^3 + 16)*log(x)^2 + x*e^6 - 8*x*e^3 - 6*(e^3 - 4)*e^(x^2 + 2) - 2*((e^3 - 4)*e^(x^2 + 2) - 3*e^6 + 24*e^3 - 48)*log(x) + 16*x + e^(2*x^2 + 4))/(e^6 - 8*e^3 + 16)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x + \frac {\left (- 2 e^{6} \log {\left (x \right )} - 32 \log {\left (x \right )} + 16 e^{3} \log {\left (x \right )} - 6 e^{6} - 96 + 48 e^{3}\right ) e^{x^{2} + 2} + \left (-4 + e^{3}\right ) e^{2 x^{2} + 4}}{- 12 e^{6} - 64 + 48 e^{3} + e^{9}} + \log {\left (x \right )}^{2} + 6 \log {\left (x \right )} \] Input:
integrate((((-4*x**2*exp(3)+16*x**2)*exp(x**2+2)+2*exp(3)**2-16*exp(3)+32) *ln(x)+4*x**2*exp(x**2+2)**2+((-12*x**2-2)*exp(3)+48*x**2+8)*exp(x**2+2)+( 6+x)*exp(3)**2+(-8*x-48)*exp(3)+16*x+96)/(x*exp(3)**2-8*x*exp(3)+16*x),x)
Output:
x + ((-2*exp(6)*log(x) - 32*log(x) + 16*exp(3)*log(x) - 6*exp(6) - 96 + 48 *exp(3))*exp(x**2 + 2) + (-4 + exp(3))*exp(2*x**2 + 4))/(-12*exp(6) - 64 + 48*exp(3) + exp(9)) + log(x)**2 + 6*log(x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 457, normalized size of antiderivative = 19.04 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=-{\left (\frac {2 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {\log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16}\right )} e^{6} + 8 \, {\left (\frac {2 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {\log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16}\right )} e^{3} + \frac {2 \, e^{6} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {16 \, e^{3} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right ) \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {x e^{6}}{e^{6} - 8 \, e^{3} + 16} + \frac {{\rm Ei}\left (x^{2}\right ) e^{5}}{e^{6} - 8 \, e^{3} + 16} - \frac {8 \, x e^{3}}{e^{6} - 8 \, e^{3} + 16} - \frac {4 \, {\rm Ei}\left (x^{2}\right ) e^{2}}{e^{6} - 8 \, e^{3} + 16} + \frac {6 \, e^{6} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {48 \, e^{3} \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} - \frac {2 \, e^{\left (x^{2} + 5\right )} \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {8 \, e^{\left (x^{2} + 2\right )} \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} + \frac {16 \, \log \left (x\right )^{2}}{e^{6} - 8 \, e^{3} + 16} + \frac {16 \, x}{e^{6} - 8 \, e^{3} + 16} - \frac {{\rm Ei}\left (x^{2}\right ) e^{5}}{{\left (e^{3} - 4\right )}^{2}} + \frac {4 \, {\rm Ei}\left (x^{2}\right ) e^{2}}{{\left (e^{3} - 4\right )}^{2}} + \frac {e^{\left (2 \, x^{2} + 4\right )}}{e^{6} - 8 \, e^{3} + 16} - \frac {6 \, e^{\left (x^{2} + 5\right )}}{e^{6} - 8 \, e^{3} + 16} + \frac {24 \, e^{\left (x^{2} + 2\right )}}{e^{6} - 8 \, e^{3} + 16} + \frac {96 \, \log \left (x e^{6} - 8 \, x e^{3} + 16 \, x\right )}{e^{6} - 8 \, e^{3} + 16} \] Input:
integrate((((-4*x^2*exp(3)+16*x^2)*exp(x^2+2)+2*exp(3)^2-16*exp(3)+32)*log (x)+4*x^2*exp(x^2+2)^2+((-12*x^2-2)*exp(3)+48*x^2+8)*exp(x^2+2)+(6+x)*exp( 3)^2+(-8*x-48)*exp(3)+16*x+96)/(x*exp(3)^2-8*x*exp(3)+16*x),x, algorithm=" maxima")
Output:
-(2*log(x*e^6 - 8*x*e^3 + 16*x)*log(x)/(e^6 - 8*e^3 + 16) - log(x)^2/(e^6 - 8*e^3 + 16))*e^6 + 8*(2*log(x*e^6 - 8*x*e^3 + 16*x)*log(x)/(e^6 - 8*e^3 + 16) - log(x)^2/(e^6 - 8*e^3 + 16))*e^3 + 2*e^6*log(x*e^6 - 8*x*e^3 + 16* x)*log(x)/(e^6 - 8*e^3 + 16) - 16*e^3*log(x*e^6 - 8*x*e^3 + 16*x)*log(x)/( e^6 - 8*e^3 + 16) + x*e^6/(e^6 - 8*e^3 + 16) + Ei(x^2)*e^5/(e^6 - 8*e^3 + 16) - 8*x*e^3/(e^6 - 8*e^3 + 16) - 4*Ei(x^2)*e^2/(e^6 - 8*e^3 + 16) + 6*e^ 6*log(x*e^6 - 8*x*e^3 + 16*x)/(e^6 - 8*e^3 + 16) - 48*e^3*log(x*e^6 - 8*x* e^3 + 16*x)/(e^6 - 8*e^3 + 16) - 2*e^(x^2 + 5)*log(x)/(e^6 - 8*e^3 + 16) + 8*e^(x^2 + 2)*log(x)/(e^6 - 8*e^3 + 16) + 16*log(x)^2/(e^6 - 8*e^3 + 16) + 16*x/(e^6 - 8*e^3 + 16) - Ei(x^2)*e^5/(e^3 - 4)^2 + 4*Ei(x^2)*e^2/(e^3 - 4)^2 + e^(2*x^2 + 4)/(e^6 - 8*e^3 + 16) - 6*e^(x^2 + 5)/(e^6 - 8*e^3 + 16 ) + 24*e^(x^2 + 2)/(e^6 - 8*e^3 + 16) + 96*log(x*e^6 - 8*x*e^3 + 16*x)/(e^ 6 - 8*e^3 + 16)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.38 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{6} \log \left (x\right )^{2} - 8 \, e^{3} \log \left (x\right )^{2} + x e^{6} - 8 \, x e^{3} + 6 \, e^{6} \log \left (x\right ) - 48 \, e^{3} \log \left (x\right ) - 2 \, e^{\left (x^{2} + 5\right )} \log \left (x\right ) + 8 \, e^{\left (x^{2} + 2\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} + 16 \, x + e^{\left (2 \, x^{2} + 4\right )} - 6 \, e^{\left (x^{2} + 5\right )} + 24 \, e^{\left (x^{2} + 2\right )} + 96 \, \log \left (x\right )}{e^{6} - 8 \, e^{3} + 16} \] Input:
integrate((((-4*x^2*exp(3)+16*x^2)*exp(x^2+2)+2*exp(3)^2-16*exp(3)+32)*log (x)+4*x^2*exp(x^2+2)^2+((-12*x^2-2)*exp(3)+48*x^2+8)*exp(x^2+2)+(6+x)*exp( 3)^2+(-8*x-48)*exp(3)+16*x+96)/(x*exp(3)^2-8*x*exp(3)+16*x),x, algorithm=" giac")
Output:
(e^6*log(x)^2 - 8*e^3*log(x)^2 + x*e^6 - 8*x*e^3 + 6*e^6*log(x) - 48*e^3*l og(x) - 2*e^(x^2 + 5)*log(x) + 8*e^(x^2 + 2)*log(x) + 16*log(x)^2 + 16*x + e^(2*x^2 + 4) - 6*e^(x^2 + 5) + 24*e^(x^2 + 2) + 96*log(x))/(e^6 - 8*e^3 + 16)
Time = 2.94 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=x+6\,\ln \left (x\right )+{\ln \left (x\right )}^2-{\mathrm {e}}^{x^2+2}\,\left (\frac {6}{{\mathrm {e}}^3-4}+\frac {2\,\ln \left (x\right )}{{\mathrm {e}}^3-4}\right )+\frac {{\mathrm {e}}^{2\,x^2+4}}{{\left ({\mathrm {e}}^3-4\right )}^2} \] Input:
int((16*x + 4*x^2*exp(2*x^2 + 4) - log(x)*(16*exp(3) - 2*exp(6) + exp(x^2 + 2)*(4*x^2*exp(3) - 16*x^2) - 32) + exp(6)*(x + 6) + exp(x^2 + 2)*(48*x^2 - exp(3)*(12*x^2 + 2) + 8) - exp(3)*(8*x + 48) + 96)/(16*x - 8*x*exp(3) + x*exp(6)),x)
Output:
x + 6*log(x) + log(x)^2 - exp(x^2 + 2)*(6/(exp(3) - 4) + (2*log(x))/(exp(3 ) - 4)) + exp(2*x^2 + 4)/(exp(3) - 4)^2
Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17 \[ \int \frac {96+e^3 (-48-8 x)+16 x+4 e^{4+2 x^2} x^2+e^6 (6+x)+e^{2+x^2} \left (8+48 x^2+e^3 \left (-2-12 x^2\right )\right )+\left (32-16 e^3+2 e^6+e^{2+x^2} \left (16 x^2-4 e^3 x^2\right )\right ) \log (x)}{16 x-8 e^3 x+e^6 x} \, dx=\frac {e^{2 x^{2}} e^{4}-2 e^{x^{2}} \mathrm {log}\left (x \right ) e^{5}+8 e^{x^{2}} \mathrm {log}\left (x \right ) e^{2}-6 e^{x^{2}} e^{5}+24 e^{x^{2}} e^{2}+\mathrm {log}\left (x \right )^{2} e^{6}-8 \mathrm {log}\left (x \right )^{2} e^{3}+16 \mathrm {log}\left (x \right )^{2}+6 \,\mathrm {log}\left (x \right ) e^{6}-48 \,\mathrm {log}\left (x \right ) e^{3}+96 \,\mathrm {log}\left (x \right )+e^{6} x -8 e^{3} x +16 x}{e^{6}-8 e^{3}+16} \] Input:
int((((-4*x^2*exp(3)+16*x^2)*exp(x^2+2)+2*exp(3)^2-16*exp(3)+32)*log(x)+4* x^2*exp(x^2+2)^2+((-12*x^2-2)*exp(3)+48*x^2+8)*exp(x^2+2)+(6+x)*exp(3)^2+( -8*x-48)*exp(3)+16*x+96)/(x*exp(3)^2-8*x*exp(3)+16*x),x)
Output:
(e**(2*x**2)*e**4 - 2*e**(x**2)*log(x)*e**5 + 8*e**(x**2)*log(x)*e**2 - 6* e**(x**2)*e**5 + 24*e**(x**2)*e**2 + log(x)**2*e**6 - 8*log(x)**2*e**3 + 1 6*log(x)**2 + 6*log(x)*e**6 - 48*log(x)*e**3 + 96*log(x) + e**6*x - 8*e**3 *x + 16*x)/(e**6 - 8*e**3 + 16)