\(\int \frac {(-2 x^2+2 x \log (x)) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x)))+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} (2 x-2 x^2+(2 x^2-2 x \log (x)) \log (x-\log (x))+(2 x+2 x^2+(-2-2 x) \log (x)) \log ^2(x-\log (x)))}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx\) [7553]

Optimal result

Integrand size = 187, antiderivative size = 22 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(22)=44\).

Time = 1.98 (sec) , antiderivative size = 155, normalized size of antiderivative = 7.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6874, 6838, 2326} \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^2-\frac {2 e^{x+\frac {x}{\log (x-\log (x))}+1} \left (-x^2+x^2 \log ^2(x-\log (x))+x^2 \log (x-\log (x))+x-x \log (x) \log ^2(x-\log (x))-x \log (x) \log (x-\log (x))\right )}{(x-\log (x)) \left (-\frac {\left (1-\frac {1}{x}\right ) x}{(x-\log (x)) \log ^2(x-\log (x))}+\frac {1}{\log (x-\log (x))}+1\right ) \log ^2(x-\log (x))}+e^{2 x+\frac {2 x}{\log (x-\log (x))}+2} \]

[In]

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

[Out]

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

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \]

[In]

Integrate[((-2*x^2 + 2*x*Log[x])*Log[x - Log[x]]^2 + E^((2*(x + (1 + x)*Log[x - Log[x]]))/Log[x - Log[x]])*(-2
+ 2*x + (-2*x + 2*Log[x])*Log[x - Log[x]] + (-2*x + 2*Log[x])*Log[x - Log[x]]^2) + E^((x + (1 + x)*Log[x - Lo
g[x]])/Log[x - Log[x]])*(2*x - 2*x^2 + (2*x^2 - 2*x*Log[x])*Log[x - Log[x]] + (2*x + 2*x^2 + (-2 - 2*x)*Log[x]
)*Log[x - Log[x]]^2))/((-x + Log[x])*Log[x - Log[x]]^2),x]

[Out]

(E^(1 + x + x/Log[x - Log[x]]) - x)^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(21)=42\).

Time = 9.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64

method	result	size
parallelrisch	\(x^{2}-2 x \,{\mathrm e}^{\frac {\left (1+x \right ) \ln \left (x -\ln \left (x \right )\right )+x}{\ln \left (x -\ln \left (x \right )\right )}}+{\mathrm e}^{\frac {2 \left (1+x \right ) \ln \left (x -\ln \left (x \right )\right )+2 x}{\ln \left (x -\ln \left (x \right )\right )}}\)	\(58\)
risch	\(x^{2}-2 x \,{\mathrm e}^{\frac {x \ln \left (x -\ln \left (x \right )\right )+\ln \left (x -\ln \left (x \right )\right )+x}{\ln \left (x -\ln \left (x \right )\right )}}+{\mathrm e}^{\frac {2 x \ln \left (x -\ln \left (x \right )\right )+2 \ln \left (x -\ln \left (x \right )\right )+2 x}{\ln \left (x -\ln \left (x \right )\right )}}\)	\(67\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 \, x e^{\left (\frac {{\left (x + 1\right )} \log \left (x - \log \left (x\right )\right ) + x}{\log \left (x - \log \left (x\right )\right )}\right )} + e^{\left (\frac {2 \, {\left ({\left (x + 1\right )} \log \left (x - \log \left (x\right )\right ) + x\right )}}{\log \left (x - \log \left (x\right )\right )}\right )} \]

[In]

integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2*x-2)*exp(((1+x)*log(x-log(x))+x)/log
(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((
1+x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(log(x)-x)/log(x-log(x))^2,x, algorit
hm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 1.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 x e^{\frac {x + \left (x + 1\right ) \log {\left (x - \log {\left (x \right )} \right )}}{\log {\left (x - \log {\left (x \right )} \right )}}} + e^{\frac {2 \left (x + \left (x + 1\right ) \log {\left (x - \log {\left (x \right )} \right )}\right )}{\log {\left (x - \log {\left (x \right )} \right )}}} \]

[In]

integrate((((2*ln(x)-2*x)*ln(x-ln(x))**2+(2*ln(x)-2*x)*ln(x-ln(x))+2*x-2)*exp(((1+x)*ln(x-ln(x))+x)/ln(x-ln(x)
))**2+(((-2-2*x)*ln(x)+2*x**2+2*x)*ln(x-ln(x))**2+(-2*x*ln(x)+2*x**2)*ln(x-ln(x))-2*x**2+2*x)*exp(((1+x)*ln(x-
ln(x))+x)/ln(x-ln(x)))+(2*x*ln(x)-2*x**2)*ln(x-ln(x))**2)/(ln(x)-x)/ln(x-ln(x))**2,x)

[Out]

x**2 - 2*x*exp((x + (x + 1)*log(x - log(x)))/log(x - log(x))) + exp(2*(x + (x + 1)*log(x - log(x)))/log(x - lo
g(x)))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2*x-2)*exp(((1+x)*log(x-log(x))+x)/log
(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((
1+x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(log(x)-x)/log(x-log(x))^2,x, algorit
hm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).

Time = 2.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 \, x e^{\left (\frac {x \log \left (x - \log \left (x\right )\right ) + x + \log \left (x - \log \left (x\right )\right )}{\log \left (x - \log \left (x\right )\right )}\right )} + e^{\left (2 \, x + \frac {2 \, x}{\log \left (x - \log \left (x\right )\right )} + 2\right )} \]

[In]

integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2*x-2)*exp(((1+x)*log(x-log(x))+x)/log
(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((
1+x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(log(x)-x)/log(x-log(x))^2,x, algorit
hm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^2+{\mathrm {e}}^{\frac {2\,x}{\ln \left (x-\ln \left (x\right )\right )}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2-2\,x\,{\mathrm {e}}^{\frac {x}{\ln \left (x-\ln \left (x\right )\right )}}\,\mathrm {e}\,{\mathrm {e}}^x \]

[In]

int(-(exp((x + log(x - log(x))*(x + 1))/log(x - log(x)))*(2*x + log(x - log(x))^2*(2*x - log(x)*(2*x + 2) + 2*
x^2) - log(x - log(x))*(2*x*log(x) - 2*x^2) - 2*x^2) + log(x - log(x))^2*(2*x*log(x) - 2*x^2) - exp((2*(x + lo
g(x - log(x))*(x + 1)))/log(x - log(x)))*(log(x - log(x))*(2*x - 2*log(x)) - 2*x + log(x - log(x))^2*(2*x - 2*
log(x)) + 2))/(log(x - log(x))^2*(x - log(x))),x)

[Out]

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