Integrand size = 78, antiderivative size = 26 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\frac {4}{3 e^{2/3}}-2 x+\frac {-2+\log \left (\log \left (x^2\right )\right )}{x}} \] Output:
exp(4/3*exp(-2/3)+(ln(ln(x^2))-2)/x-2*x)
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \] Input:
Integrate[(E^((4*x + E^(2/3)*(-6 - 6*x^2) + 3*E^(2/3)*Log[Log[x^2]])/(3*E^ (2/3)*x))*(2 + (2 - 2*x^2)*Log[x^2] - Log[x^2]*Log[Log[x^2]]))/(x^2*Log[x^ 2]),x]
Output:
E^(4/(3*E^(2/3)) - 2/x - 2*x)*Log[x^2]^x^(-1)
Time = 0.77 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {7257}
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.
\(\displaystyle \int \frac {\left (\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )+2\right ) \exp \left (\frac {e^{2/3} \left (-6 x^2-6\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )+4 x}{3 e^{2/3} x}\right )}{x^2 \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (\frac {2 \left (2 x-3 e^{2/3} \left (x^2+1\right )\right )}{3 e^{2/3} x}\right ) \sqrt [x]{\log \left (x^2\right )}\) |
Input:
Int[(E^((4*x + E^(2/3)*(-6 - 6*x^2) + 3*E^(2/3)*Log[Log[x^2]])/(3*E^(2/3)* x))*(2 + (2 - 2*x^2)*Log[x^2] - Log[x^2]*Log[Log[x^2]]))/(x^2*Log[x^2]),x]
Output:
E^((2*(2*x - 3*E^(2/3)*(1 + x^2)))/(3*E^(2/3)*x))*Log[x^2]^x^(-1)
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 0.54 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (3 \,{\mathrm e}^{\frac {2}{3}} \ln \left (\ln \left (x^{2}\right )\right )+\left (-6 x^{2}-6\right ) {\mathrm e}^{\frac {2}{3}}+4 x \right ) {\mathrm e}^{-\frac {2}{3}}}{3 x}}\) | \(34\) |
Input:
int((-ln(x^2)*ln(ln(x^2))+(-2*x^2+2)*ln(x^2)+2)*exp(1/3*(3*exp(2/3)*ln(ln( x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/ln(x^2),x,method=_RETURNVER BOSE)
Output:
exp(1/3*(3*exp(2/3)*ln(ln(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\left (-\frac {{\left (6 \, {\left (x^{2} + 1\right )} e^{\frac {2}{3}} - 3 \, e^{\frac {2}{3}} \log \left (\log \left (x^{2}\right )\right ) - 4 \, x\right )} e^{\left (-\frac {2}{3}\right )}}{3 \, x}\right )} \] Input:
integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2 /3)*log(log(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/log(x^2),x, alg orithm="fricas")
Output:
e^(-1/3*(6*(x^2 + 1)*e^(2/3) - 3*e^(2/3)*log(log(x^2)) - 4*x)*e^(-2/3)/x)
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\frac {\frac {4 x}{3} + \frac {\left (- 6 x^{2} - 6\right ) e^{\frac {2}{3}}}{3} + e^{\frac {2}{3}} \log {\left (\log {\left (x^{2} \right )} \right )}}{x e^{\frac {2}{3}}}} \] Input:
integrate((-ln(x**2)*ln(ln(x**2))+(-2*x**2+2)*ln(x**2)+2)*exp(1/3*(3*exp(2 /3)*ln(ln(x**2))+(-6*x**2-6)*exp(2/3)+4*x)/x/exp(2/3))/x**2/ln(x**2),x)
Output:
exp((4*x/3 + (-6*x**2 - 6)*exp(2/3)/3 + exp(2/3)*log(log(x**2)))*exp(-2/3) /x)
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\left (-2 \, x + \frac {\log \left (2\right )}{x} + \frac {\log \left (\log \left (x\right )\right )}{x} - \frac {2}{x} + \frac {4}{3} \, e^{\left (-\frac {2}{3}\right )}\right )} \] Input:
integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2 /3)*log(log(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/log(x^2),x, alg orithm="maxima")
Output:
e^(-2*x + log(2)/x + log(log(x))/x - 2/x + 4/3*e^(-2/3))
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=e^{\left (-\frac {6 \, x^{2} - 4 \, x e^{\left (-\frac {2}{3}\right )} - 3 \, \log \left (\log \left (x^{2}\right )\right ) + 6}{3 \, x}\right )} \] Input:
integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2 /3)*log(log(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/log(x^2),x, alg orithm="giac")
Output:
e^(-1/3*(6*x^2 - 4*x*e^(-2/3) - 3*log(log(x^2)) + 6)/x)
Time = 2.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx={\ln \left (x^2\right )}^{1/x}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-\frac {2}{3}}}{3}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-\frac {2}{x}} \] Input:
int(-(exp((exp(-2/3)*((4*x)/3 - (exp(2/3)*(6*x^2 + 6))/3 + exp(2/3)*log(lo g(x^2))))/x)*(log(x^2)*(2*x^2 - 2) + log(x^2)*log(log(x^2)) - 2))/(x^2*log (x^2)),x)
Output:
log(x^2)^(1/x)*exp((4*exp(-2/3))/3)*exp(-2*x)*exp(-2/x)
\[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx=\int \frac {\left (-\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )+\left (-2 x^{2}+2\right ) \mathrm {log}\left (x^{2}\right )+2\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}^{\frac {2}{3}} \mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )+\left (-6 x^{2}-6\right ) {\mathrm e}^{\frac {2}{3}}+4 x}{3 x \,{\mathrm e}^{\frac {2}{3}}}}}{x^{2} \mathrm {log}\left (x^{2}\right )}d x \] Input:
int((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2/3)*lo g(log(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/log(x^2),x)
Output:
int((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2/3)*lo g(log(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp(2/3))/x^2/log(x^2),x)