\(\int \frac {e^{\frac {4 e^{2 \log ^2(x^2)} (1-x)}{x^2}+2 \log ^2(x^2)} (-8+4 x+(32-32 x) \log (x^2))}{x^3} \, dx\) [7410]

Optimal result

Integrand size = 50, antiderivative size = 22 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}} \]

[Out]

exp((1-x)*exp(ln(x^2)^2)^2/x^2)^4

Rubi [F]

\[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=\int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx \]

[In]

Int[(E^((4*E^(2*Log[x^2]^2)*(1 - x))/x^2 + 2*Log[x^2]^2)*(-8 + 4*x + (32 - 32*x)*Log[x^2]))/x^3,x]

[Out]

-8*Defer[Int][E^((4*E^(2*Log[x^2]^2)*(1 - x))/x^2 + 2*Log[x^2]^2)/x^3, x] + 4*Defer[Int][E^((4*E^(2*Log[x^2]^2
)*(1 - x))/x^2 + 2*Log[x^2]^2)/x^2, x] + 32*Defer[Int][(E^((4*E^(2*Log[x^2]^2)*(1 - x))/x^2 + 2*Log[x^2]^2)*Lo
g[x^2])/x^3, x] - 32*Defer[Int][(E^((4*E^(2*Log[x^2]^2)*(1 - x))/x^2 + 2*Log[x^2]^2)*Log[x^2])/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 \exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) (-2+x)}{x^3}-\frac {32 \exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) (-1+x) \log \left (x^2\right )}{x^3}\right ) \, dx \\ & = 4 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) (-2+x)}{x^3} \, dx-32 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) (-1+x) \log \left (x^2\right )}{x^3} \, dx \\ & = 4 \int \left (-\frac {2 \exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right )}{x^3}+\frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right )}{x^2}\right ) \, dx-32 \int \left (-\frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^3}+\frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2}\right ) \, dx \\ & = 4 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right )}{x^2} \, dx-8 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right )}{x^3} \, dx+32 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^3} \, dx-32 \int \frac {\exp \left (\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=e^{-\frac {4 e^{2 \log ^2\left (x^2\right )} (-1+x)}{x^2}} \]

[In]

Integrate[(E^((4*E^(2*Log[x^2]^2)*(1 - x))/x^2 + 2*Log[x^2]^2)*(-8 + 4*x + (32 - 32*x)*Log[x^2]))/x^3,x]

[Out]

E^((-4*E^(2*Log[x^2]^2)*(-1 + x))/x^2)

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
risch	\({\mathrm e}^{-\frac {4 \left (-1+x \right ) {\mathrm e}^{2 \ln \left (x^{2}\right )^{2}}}{x^{2}}}\)	\(19\)
parallelrisch	\({\mathrm e}^{\frac {4 \left (1-x \right ) {\mathrm e}^{2 \ln \left (x^{2}\right )^{2}}}{x^{2}}}\)	\(22\)

[In]

int(((-32*x+32)*ln(x^2)+4*x-8)*exp(ln(x^2)^2)^2*exp((1-x)*exp(ln(x^2)^2)^2/x^2)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(-4*(-1+x)*exp(2*ln(x^2)^2)/x^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=e^{\left (-2 \, \log \left (x^{2}\right )^{2} + \frac {2 \, {\left (x^{2} \log \left (x^{2}\right )^{2} - 2 \, {\left (x - 1\right )} e^{\left (2 \, \log \left (x^{2}\right )^{2}\right )}\right )}}{x^{2}}\right )} \]

[In]

integrate(((-32*x+32)*log(x^2)+4*x-8)*exp(log(x^2)^2)^2*exp((1-x)*exp(log(x^2)^2)^2/x^2)^4/x^3,x, algorithm="f
ricas")

[Out]

e^(-2*log(x^2)^2 + 2*(x^2*log(x^2)^2 - 2*(x - 1)*e^(2*log(x^2)^2))/x^2)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=e^{\frac {4 \cdot \left (1 - x\right ) e^{2 \log {\left (x^{2} \right )}^{2}}}{x^{2}}} \]

[In]

integrate(((-32*x+32)*ln(x**2)+4*x-8)*exp(ln(x**2)**2)**2*exp((1-x)*exp(ln(x**2)**2)**2/x**2)**4/x**3,x)

[Out]

exp(4*(1 - x)*exp(2*log(x**2)**2)/x**2)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=e^{\left (-\frac {4 \, e^{\left (8 \, \log \left (x\right )^{2}\right )}}{x} + \frac {4 \, e^{\left (8 \, \log \left (x\right )^{2}\right )}}{x^{2}}\right )} \]

[In]

integrate(((-32*x+32)*log(x^2)+4*x-8)*exp(log(x^2)^2)^2*exp((1-x)*exp(log(x^2)^2)^2/x^2)^4/x^3,x, algorithm="m
axima")

[Out]

e^(-4*e^(8*log(x)^2)/x + 4*e^(8*log(x)^2)/x^2)

Giac [F]

\[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx=\int { -\frac {4 \, {\left (8 \, {\left (x - 1\right )} \log \left (x^{2}\right ) - x + 2\right )} e^{\left (2 \, \log \left (x^{2}\right )^{2} - \frac {4 \, {\left (x - 1\right )} e^{\left (2 \, \log \left (x^{2}\right )^{2}\right )}}{x^{2}}\right )}}{x^{3}} \,d x } \]

[In]

integrate(((-32*x+32)*log(x^2)+4*x-8)*exp(log(x^2)^2)^2*exp((1-x)*exp(log(x^2)^2)^2/x^2)^4/x^3,x, algorithm="g
iac")

[Out]

integrate(-4*(8*(x - 1)*log(x^2) - x + 2)*e^(2*log(x^2)^2 - 4*(x - 1)*e^(2*log(x^2)^2)/x^2)/x^3, x)

Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {4 e^{2 \log ^2\left (x^2\right )} (1-x)}{x^2}+2 \log ^2\left (x^2\right )} \left (-8+4 x+(32-32 x) \log \left (x^2\right )\right )}{x^3} \, dx={\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{2\,{\ln \left (x^2\right )}^2}}{x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{2\,{\ln \left (x^2\right )}^2}}{x^2}} \]

[In]

int(-(exp(2*log(x^2)^2)*exp(-(4*exp(2*log(x^2)^2)*(x - 1))/x^2)*(log(x^2)*(32*x - 32) - 4*x + 8))/x^3,x)

[Out]

exp(-(4*exp(2*log(x^2)^2))/x)*exp((4*exp(2*log(x^2)^2))/x^2)