Integrand size = 78, antiderivative size = 28 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=\frac {x}{2}+\log (x)+\left (4 e^{x^2}-x\right )^2 \log ^2\left (x^2\right ) \]
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=\frac {1}{2} \left (x+2 \log (x)+2 \left (-4 e^{x^2}+x\right )^2 \log ^2\left (x^2\right )\right ) \]
Integrate[(2 + x + (128*E^(2*x^2) - 64*E^x^2*x + 8*x^2)*Log[x^2] + (4*x^2 + 128*E^(2*x^2)*x^2 + E^x^2*(-16*x - 32*x^3))*Log[x^2]^2)/(2*x),x]
Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {27, 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.
\(\displaystyle \int \frac {\left (8 x^2-64 e^{x^2} x+128 e^{2 x^2}\right ) \log \left (x^2\right )+\left (128 e^{2 x^2} x^2+4 x^2+e^{x^2} \left (-32 x^3-16 x\right )\right ) \log ^2\left (x^2\right )+x+2}{2 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {4 \left (32 e^{2 x^2} x^2+x^2-4 e^{x^2} \left (2 x^3+x\right )\right ) \log ^2\left (x^2\right )+8 \left (x^2-8 e^{x^2} x+16 e^{2 x^2}\right ) \log \left (x^2\right )+x+2}{x}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{2} \int \left (\frac {128 e^{2 x^2} \log \left (x^2\right ) \left (\log \left (x^2\right ) x^2+1\right )}{x}-16 e^{x^2} \log \left (x^2\right ) \left (2 \log \left (x^2\right ) x^2+\log \left (x^2\right )+4\right )+\frac {4 \log ^2\left (x^2\right ) x^2+8 \log \left (x^2\right ) x^2+x+2}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (32 e^{2 x^2} \log ^2\left (x^2\right )+2 x^2 \log ^2\left (x^2\right )-16 e^{x^2} x \log ^2\left (x^2\right )+x+2 \log (x)\right )\) |
Int[(2 + x + (128*E^(2*x^2) - 64*E^x^2*x + 8*x^2)*Log[x^2] + (4*x^2 + 128* E^(2*x^2)*x^2 + E^x^2*(-16*x - 32*x^3))*Log[x^2]^2)/(2*x),x]
3.7.64.3.1 Defintions of rubi rules used
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]]
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57
method | result | size |
parallelrisch | \(x^{2} \ln \left (x^{2}\right )^{2}-8 \,{\mathrm e}^{x^{2}} \ln \left (x^{2}\right )^{2} x +16 \,{\mathrm e}^{2 x^{2}} \ln \left (x^{2}\right )^{2}+\ln \left (x \right )+\frac {x}{2}\) | \(44\) |
default | \(\ln \left (x \right )+\frac {x}{2}+64 \,{\mathrm e}^{2 x^{2}} \ln \left (x \right )^{2}+64 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) {\mathrm e}^{2 x^{2}} \ln \left (x \right )+16 {\left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )}^{2} {\mathrm e}^{2 x^{2}}-32 x \,{\mathrm e}^{x^{2}} \ln \left (x \right )^{2}-8 {\left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )}^{2} x \,{\mathrm e}^{x^{2}}-32 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) x \,{\mathrm e}^{x^{2}} \ln \left (x \right )+x^{2} \ln \left (x^{2}\right )^{2}\) | \(114\) |
parts | \(\ln \left (x \right )+\frac {x}{2}+64 \,{\mathrm e}^{2 x^{2}} \ln \left (x \right )^{2}+64 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) {\mathrm e}^{2 x^{2}} \ln \left (x \right )+16 {\left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )}^{2} {\mathrm e}^{2 x^{2}}-32 x \,{\mathrm e}^{x^{2}} \ln \left (x \right )^{2}-8 {\left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )}^{2} x \,{\mathrm e}^{x^{2}}-32 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) x \,{\mathrm e}^{x^{2}} \ln \left (x \right )+x^{2} \ln \left (x^{2}\right )^{2}\) | \(114\) |
risch | \(\frac {\left (128 \,{\mathrm e}^{2 x^{2}}-64 \,{\mathrm e}^{x^{2}} x +8 x^{2}\right ) \ln \left (x \right )^{2}}{2}-2 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (x^{2} \operatorname {csgn}\left (i x \right )^{2}-2 x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )+x^{2} \operatorname {csgn}\left (i x^{2}\right )^{2}-8 x \operatorname {csgn}\left (i x \right )^{2} {\mathrm e}^{x^{2}}+16 x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x^{2}}-8 x \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x^{2}}+16 \operatorname {csgn}\left (i x \right )^{2} {\mathrm e}^{2 x^{2}}-32 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{2 x^{2}}+16 \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{2 x^{2}}\right ) \ln \left (x \right )-\frac {\pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}}{4}+\pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-\frac {3 \pi ^{2} x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}}{2}+\pi ^{2} x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-\frac {\pi ^{2} x^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}}{4}+\frac {x}{2}+\ln \left (x \right )-4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{2 x^{2}}+16 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{2 x^{2}}-24 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} {\mathrm e}^{2 x^{2}}+16 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5} {\mathrm e}^{2 x^{2}}-4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} {\mathrm e}^{2 x^{2}}+2 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x^{2}}-8 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x^{2}}+12 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} {\mathrm e}^{x^{2}}-8 \pi ^{2} x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5} {\mathrm e}^{x^{2}}+2 \pi ^{2} x \operatorname {csgn}\left (i x^{2}\right )^{6} {\mathrm e}^{x^{2}}\) | \(546\) |
int(1/2*((128*x^2*exp(x^2)^2+(-32*x^3-16*x)*exp(x^2)+4*x^2)*ln(x^2)^2+(128 *exp(x^2)^2-64*exp(x^2)*x+8*x^2)*ln(x^2)+2+x)/x,x,method=_RETURNVERBOSE)
Time = 0.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx={\left (x^{2} - 8 \, x e^{\left (x^{2}\right )} + 16 \, e^{\left (2 \, x^{2}\right )}\right )} \log \left (x^{2}\right )^{2} + \frac {1}{2} \, x + \frac {1}{2} \, \log \left (x^{2}\right ) \]
integrate(1/2*((128*x^2*exp(x^2)^2+(-32*x^3-16*x)*exp(x^2)+4*x^2)*log(x^2) ^2+(128*exp(x^2)^2-64*exp(x^2)*x+8*x^2)*log(x^2)+2+x)/x,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=x^{2} \log {\left (x^{2} \right )}^{2} - 8 x e^{x^{2}} \log {\left (x^{2} \right )}^{2} + \frac {x}{2} + 16 e^{2 x^{2}} \log {\left (x^{2} \right )}^{2} + \log {\left (x \right )} \]
integrate(1/2*((128*x**2*exp(x**2)**2+(-32*x**3-16*x)*exp(x**2)+4*x**2)*ln (x**2)**2+(128*exp(x**2)**2-64*exp(x**2)*x+8*x**2)*ln(x**2)+2+x)/x,x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=4 \, x^{2} \log \left (x\right )^{2} - 32 \, x e^{\left (x^{2}\right )} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x^{2}\right ) - 4 \, x^{2} \log \left (x\right ) + 64 \, e^{\left (2 \, x^{2}\right )} \log \left (x\right )^{2} + \frac {1}{2} \, x + \log \left (x\right ) \]
integrate(1/2*((128*x^2*exp(x^2)^2+(-32*x^3-16*x)*exp(x^2)+4*x^2)*log(x^2) ^2+(128*exp(x^2)^2-64*exp(x^2)*x+8*x^2)*log(x^2)+2+x)/x,x, algorithm=\
4*x^2*log(x)^2 - 32*x*e^(x^2)*log(x)^2 + 2*x^2*log(x^2) - 4*x^2*log(x) + 6 4*e^(2*x^2)*log(x)^2 + 1/2*x + log(x)
Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=x^{2} \log \left (x^{2}\right )^{2} - 8 \, x e^{\left (x^{2}\right )} \log \left (x^{2}\right )^{2} + 16 \, e^{\left (2 \, x^{2}\right )} \log \left (x^{2}\right )^{2} + \frac {1}{2} \, x + \log \left (x\right ) \]
integrate(1/2*((128*x^2*exp(x^2)^2+(-32*x^3-16*x)*exp(x^2)+4*x^2)*log(x^2) ^2+(128*exp(x^2)^2-64*exp(x^2)*x+8*x^2)*log(x^2)+2+x)/x,x, algorithm=\
Time = 13.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {2+x+\left (128 e^{2 x^2}-64 e^{x^2} x+8 x^2\right ) \log \left (x^2\right )+\left (4 x^2+128 e^{2 x^2} x^2+e^{x^2} \left (-16 x-32 x^3\right )\right ) \log ^2\left (x^2\right )}{2 x} \, dx=\left (16\,{\mathrm {e}}^{2\,x^2}-8\,x\,{\mathrm {e}}^{x^2}+x^2\right )\,{\ln \left (x^2\right )}^2+\frac {x}{2}+\ln \left (x\right ) \]