3.8.21 \(\int \frac {1+(x+2 e^{x^2} x^2) \log (x)+(5+2 x) \log (x) \log (4 e^{e^{x^2}+x} \log (x))+\log (x) \log (4 e^{e^{x^2}+x} \log (x)) \log (4 \log (4 e^{e^{x^2}+x} \log (x)))}{\log (x) \log (4 e^{e^{x^2}+x} \log (x))} \, dx\)

Optimal. Leaf size=22 \[ x \left (5+x+\log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 1.25, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 3, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6742, 6688, 2549} \begin {gather*} x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )+5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (x + 2*E^x^2*x^2)*Log[x] + (5 + 2*x)*Log[x]*Log[4*E^(E^x^2 + x)*Log[x]] + Log[x]*Log[4*E^(E^x^2 + x)*
Log[x]]*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]])/(Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]),x]

[Out]

5*x + x^2 + x*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]]

Rule 2549

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[
u]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\frac {1+x \log (x)+5 \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+2 x \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \frac {1+x \log (x)+5 \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+2 x \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\\ &=2 \int \frac {e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \left (5+2 x+\frac {x+\frac {1}{\log (x)}}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )\right ) \, dx\\ &=5 x+x^2+2 \int \frac {e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \frac {x+\frac {1}{\log (x)}}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right ) \, dx\\ &=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )+2 \int \frac {e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \left (\frac {x}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\frac {1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )}\right ) \, dx-\int \frac {1+x \left (1+2 e^{x^2} x\right ) \log (x)}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\\ &=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )+2 \int \frac {e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx-\int \left (\frac {2 e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\frac {1+x \log (x)}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )}\right ) \, dx+\int \frac {x}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \frac {1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\\ &=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )+\int \frac {x}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \frac {1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx-\int \frac {1+x \log (x)}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\\ &=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )-\int \left (\frac {x}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\frac {1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )}\right ) \, dx+\int \frac {x}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx+\int \frac {1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\\ &=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.30, size = 26, normalized size = 1.18 \begin {gather*} 5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + (x + 2*E^x^2*x^2)*Log[x] + (5 + 2*x)*Log[x]*Log[4*E^(E^x^2 + x)*Log[x]] + Log[x]*Log[4*E^(E^x^2
+ x)*Log[x]]*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]])/(Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]),x]

[Out]

5*x + x^2 + x*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]]

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 24, normalized size = 1.09 \begin {gather*} x^{2} + x \log \left (4 \, \log \left (4 \, e^{\left (x + e^{\left (x^{2}\right )}\right )} \log \relax (x)\right )\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp(x^2)+x)))+(5+2*x)*log(x)*log(4*log(
x)*exp(exp(x^2)+x))+(2*x^2*exp(x^2)+x)*log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="fricas")

[Out]

x^2 + x*log(4*log(4*e^(x + e^(x^2))*log(x))) + 5*x

________________________________________________________________________________________

giac [A]  time = 0.41, size = 28, normalized size = 1.27 \begin {gather*} x^{2} + 2 \, x \log \relax (2) + x \log \left (x + e^{\left (x^{2}\right )} + 2 \, \log \relax (2) + \log \left (\log \relax (x)\right )\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp(x^2)+x)))+(5+2*x)*log(x)*log(4*log(
x)*exp(exp(x^2)+x))+(2*x^2*exp(x^2)+x)*log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="giac")

[Out]

x^2 + 2*x*log(2) + x*log(x + e^(x^2) + 2*log(2) + log(log(x))) + 5*x

________________________________________________________________________________________

maple [C]  time = 0.22, size = 97, normalized size = 4.41

method	result	size
risch	\(x^{2}+x \ln \left (8 \ln \relax (2)+4 \ln \left (\ln \relax (x )\right )+4 \ln \left ({\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )-2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right ) \left (-\mathrm {csgn}\left (i \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right ) \left (-\mathrm {csgn}\left (i \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )\right )\right )+5 x\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)*ln(4*ln(x)*exp(exp(x^2)+x))*ln(4*ln(4*ln(x)*exp(exp(x^2)+x)))+(5+2*x)*ln(x)*ln(4*ln(x)*exp(exp(x^2)
+x))+(2*x^2*exp(x^2)+x)*ln(x)+1)/ln(x)/ln(4*ln(x)*exp(exp(x^2)+x)),x,method=_RETURNVERBOSE)

[Out]

x^2+x*ln(8*ln(2)+4*ln(ln(x))+4*ln(exp(exp(x^2)+x))-2*I*Pi*csgn(I*ln(x)*exp(exp(x^2)+x))*(-csgn(I*ln(x)*exp(exp
(x^2)+x))+csgn(I*ln(x)))*(-csgn(I*ln(x)*exp(exp(x^2)+x))+csgn(I*exp(exp(x^2)+x))))+5*x

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 28, normalized size = 1.27 \begin {gather*} x^{2} + 2 \, x \log \relax (2) + x \log \left (x + e^{\left (x^{2}\right )} + 2 \, \log \relax (2) + \log \left (\log \relax (x)\right )\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp(x^2)+x)))+(5+2*x)*log(x)*log(4*log(
x)*exp(exp(x^2)+x))+(2*x^2*exp(x^2)+x)*log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="maxima")

[Out]

x^2 + 2*x*log(2) + x*log(x + e^(x^2) + 2*log(2) + log(log(x))) + 5*x

________________________________________________________________________________________

mupad [B]  time = 0.88, size = 27, normalized size = 1.23 \begin {gather*} 5\,x+x\,\ln \left (4\,x+4\,{\mathrm {e}}^{x^2}+4\,\ln \left (\ln \relax (x)\right )+\ln \left (256\right )\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(x + 2*x^2*exp(x^2)) + log(4*exp(x + exp(x^2))*log(x))*log(x)*log(4*log(4*exp(x + exp(x^2))*log(x)
)) + log(4*exp(x + exp(x^2))*log(x))*log(x)*(2*x + 5) + 1)/(log(4*exp(x + exp(x^2))*log(x))*log(x)),x)

[Out]

5*x + x*log(4*x + 4*exp(x^2) + 4*log(log(x)) + log(256)) + x^2

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)*ln(4*ln(x)*exp(exp(x**2)+x))*ln(4*ln(4*ln(x)*exp(exp(x**2)+x)))+(5+2*x)*ln(x)*ln(4*ln(x)*exp(
exp(x**2)+x))+(2*x**2*exp(x**2)+x)*ln(x)+1)/ln(x)/ln(4*ln(x)*exp(exp(x**2)+x)),x)

[Out]

Timed out

________________________________________________________________________________________