Integrand size = 206, antiderivative size = 26 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{6 \left (-x+\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )} \]
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{6 \left (-x+\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )} \]
Integrate[(-13*E^(x/2) - 52*E^x^2*x + (-78 + 26*E^(x/2) + 26*E^x^2)*Log[-3 + E^(x/2) + E^x^2])/((-36*x^2 + 12*E^(x/2)*x^2 + 12*E^x^2*x^2)*Log[-3 + E ^(x/2) + E^x^2] + (72*x - 24*E^(x/2)*x - 24*E^x^2*x)*Log[-3 + E^(x/2) + E^ x^2]*Log[Log[-3 + E^(x/2) + E^x^2]] + (-36 + 12*E^(x/2) + 12*E^x^2)*Log[-3 + E^(x/2) + E^x^2]*Log[Log[-3 + E^(x/2) + E^x^2]]^2),x]
Time = 0.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {-52 e^{x^2} x+\left (26 e^{x^2}+26 e^{x/2}-78\right ) \log \left (e^{x^2}+e^{x/2}-3\right )-13 e^{x/2}}{\left (12 e^{x^2}+12 e^{x/2}-36\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \log ^2\left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )+\left (-24 e^{x^2} x-24 e^{x/2} x+72 x\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )+\left (12 e^{x^2} x^2+12 e^{x/2} x^2-36 x^2\right ) \log \left (e^{x^2}+e^{x/2}-3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {13 \left (4 e^{x^2} x-2 \left (e^{x^2}+e^{x/2}-3\right ) \log \left (e^{x^2}+e^{x/2}-3\right )+e^{x/2}\right )}{12 \left (-e^{x^2}-e^{x/2}+3\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \left (x-\log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {13}{12} \int \frac {4 e^{x^2} x+e^{x/2}+2 \left (3-e^{x/2}-e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (3-e^{x/2}-e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \left (x-\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {13}{6 \left (x-\log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )\right )}\) |
Int[(-13*E^(x/2) - 52*E^x^2*x + (-78 + 26*E^(x/2) + 26*E^x^2)*Log[-3 + E^( x/2) + E^x^2])/((-36*x^2 + 12*E^(x/2)*x^2 + 12*E^x^2*x^2)*Log[-3 + E^(x/2) + E^x^2] + (72*x - 24*E^(x/2)*x - 24*E^x^2*x)*Log[-3 + E^(x/2) + E^x^2]*L og[Log[-3 + E^(x/2) + E^x^2]] + (-36 + 12*E^(x/2) + 12*E^x^2)*Log[-3 + E^( x/2) + E^x^2]*Log[Log[-3 + E^(x/2) + E^x^2]]^2),x]
3.18.46.3.1 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 12.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {13}{6 \left (x -\ln \left (\ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )\right )}\) | \(21\) |
parallelrisch | \(-\frac {13}{6 \left (x -\ln \left (\ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )\right )}\) | \(21\) |
int(((26*exp(x^2)+26*exp(1/2*x)-78)*ln(exp(x^2)+exp(1/2*x)-3)-52*exp(x^2)* x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*ln(exp(x^2)+exp(1/2*x)-3) *ln(ln(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x)+72*x)*ln( exp(x^2)+exp(1/2*x)-3)*ln(ln(exp(x^2)+exp(1/2*x)-3))+(12*x^2*exp(x^2)+12*x ^2*exp(1/2*x)-36*x^2)*ln(exp(x^2)+exp(1/2*x)-3)),x,method=_RETURNVERBOSE)
Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \]
integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 /2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) +72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*x^2* exp(x^2)+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith m=\
Time = 6.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{- 6 x + 6 \log {\left (\log {\left (e^{\frac {x}{2}} + e^{x^{2}} - 3 \right )} \right )}} \]
integrate(((26*exp(x**2)+26*exp(1/2*x)-78)*ln(exp(x**2)+exp(1/2*x)-3)-52*e xp(x**2)*x-13*exp(1/2*x))/((12*exp(x**2)+12*exp(1/2*x)-36)*ln(exp(x**2)+ex p(1/2*x)-3)*ln(ln(exp(x**2)+exp(1/2*x)-3))**2+(-24*exp(x**2)*x-24*x*exp(1/ 2*x)+72*x)*ln(exp(x**2)+exp(1/2*x)-3)*ln(ln(exp(x**2)+exp(1/2*x)-3))+(12*x **2*exp(x**2)+12*x**2*exp(1/2*x)-36*x**2)*ln(exp(x**2)+exp(1/2*x)-3)),x)
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \]
integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 /2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) +72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*x^2* exp(x^2)+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith m=\
Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \]
integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 /2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) +72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*x^2* exp(x^2)+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith m=\
Time = 12.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6\,\left (x-\ln \left (\ln \left ({\mathrm {e}}^{x/2}+{\mathrm {e}}^{x^2}-3\right )\right )\right )} \]
int(-(13*exp(x/2) + 52*x*exp(x^2) - log(exp(x/2) + exp(x^2) - 3)*(26*exp(x /2) + 26*exp(x^2) - 78))/(log(exp(x/2) + exp(x^2) - 3)*(12*x^2*exp(x/2) + 12*x^2*exp(x^2) - 36*x^2) + log(exp(x/2) + exp(x^2) - 3)*log(log(exp(x/2) + exp(x^2) - 3))^2*(12*exp(x/2) + 12*exp(x^2) - 36) - log(exp(x/2) + exp(x ^2) - 3)*log(log(exp(x/2) + exp(x^2) - 3))*(24*x*exp(x/2) - 72*x + 24*x*ex p(x^2))),x)