3.52.24 \(\int \frac {3+3 x+(3 e^x x^2+3 e^x x \log (x)) \log (x+\log (x))}{(-16 x^2+3 e^x x^2+(-16 x+3 e^x x) \log (x)) \log (x+\log (x))+(3 x^2+3 x \log (x)) \log (x+\log (x)) \log (\log (x+\log (x)))} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {1}{45} \left (\frac {16}{3}-e^x-\log (\log (x+\log (x)))\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 16, normalized size of antiderivative = 0.73, number of steps used = 3, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6688, 12, 6684} \begin {gather*} \log \left (-3 e^x-3 \log (\log (x+\log (x)))+16\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 3*x + (3*E^x*x^2 + 3*E^x*x*Log[x])*Log[x + Log[x]])/((-16*x^2 + 3*E^x*x^2 + (-16*x + 3*E^x*x)*Log[x])
*Log[x + Log[x]] + (3*x^2 + 3*x*Log[x])*Log[x + Log[x]]*Log[Log[x + Log[x]]]),x]

[Out]

Log[16 - 3*E^x - 3*Log[Log[x + Log[x]]]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-1-x-e^x x (x+\log (x)) \log (x+\log (x))\right )}{x (x+\log (x)) \log (x+\log (x)) \left (16-3 e^x-3 \log (\log (x+\log (x)))\right )} \, dx\\ &=3 \int \frac {-1-x-e^x x (x+\log (x)) \log (x+\log (x))}{x (x+\log (x)) \log (x+\log (x)) \left (16-3 e^x-3 \log (\log (x+\log (x)))\right )} \, dx\\ &=\log \left (16-3 e^x-3 \log (\log (x+\log (x)))\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.45, size = 16, normalized size = 0.73 \begin {gather*} \log \left (-16+3 e^x+3 \log (\log (x+\log (x)))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 3*x + (3*E^x*x^2 + 3*E^x*x*Log[x])*Log[x + Log[x]])/((-16*x^2 + 3*E^x*x^2 + (-16*x + 3*E^x*x)*L
og[x])*Log[x + Log[x]] + (3*x^2 + 3*x*Log[x])*Log[x + Log[x]]*Log[Log[x + Log[x]]]),x]

[Out]

Log[-16 + 3*E^x + 3*Log[Log[x + Log[x]]]]

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 15, normalized size = 0.68 \begin {gather*} \log \left (3 \, e^{x} + 3 \, \log \left (\log \left (x + \log \relax (x)\right )\right ) - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*log(x)+3*exp(x)*x^2)*log(x+log(x))+3*x+3)/((3*x*log(x)+3*x^2)*log(x+log(x))*log(log(x+l
og(x)))+((3*exp(x)*x-16*x)*log(x)+3*exp(x)*x^2-16*x^2)*log(x+log(x))),x, algorithm="fricas")

[Out]

log(3*e^x + 3*log(log(x + log(x))) - 16)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 15, normalized size = 0.68 \begin {gather*} \log \left (-3 \, e^{x} - 3 \, \log \left (\log \left (x + \log \relax (x)\right )\right ) + 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*log(x)+3*exp(x)*x^2)*log(x+log(x))+3*x+3)/((3*x*log(x)+3*x^2)*log(x+log(x))*log(log(x+l
og(x)))+((3*exp(x)*x-16*x)*log(x)+3*exp(x)*x^2-16*x^2)*log(x+log(x))),x, algorithm="giac")

[Out]

log(-3*e^x - 3*log(log(x + log(x))) + 16)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 12, normalized size = 0.55

method	result	size
risch	\(\ln \left ({\mathrm e}^{x}+\ln \left (\ln \left (x +\ln \relax (x )\right )\right )-\frac {16}{3}\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x*exp(x)*ln(x)+3*exp(x)*x^2)*ln(x+ln(x))+3*x+3)/((3*x*ln(x)+3*x^2)*ln(x+ln(x))*ln(ln(x+ln(x)))+((3*exp
(x)*x-16*x)*ln(x)+3*exp(x)*x^2-16*x^2)*ln(x+ln(x))),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)+ln(ln(x+ln(x)))-16/3)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 11, normalized size = 0.50 \begin {gather*} \log \left (e^{x} + \log \left (\log \left (x + \log \relax (x)\right )\right ) - \frac {16}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*log(x)+3*exp(x)*x^2)*log(x+log(x))+3*x+3)/((3*x*log(x)+3*x^2)*log(x+log(x))*log(log(x+l
og(x)))+((3*exp(x)*x-16*x)*log(x)+3*exp(x)*x^2-16*x^2)*log(x+log(x))),x, algorithm="maxima")

[Out]

log(e^x + log(log(x + log(x))) - 16/3)

________________________________________________________________________________________

mupad [B]  time = 4.01, size = 11, normalized size = 0.50 \begin {gather*} \ln \left (\ln \left (\ln \left (x+\ln \relax (x)\right )\right )+{\mathrm {e}}^x-\frac {16}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + log(x + log(x))*(3*x^2*exp(x) + 3*x*exp(x)*log(x)) + 3)/(log(x + log(x))*(log(x)*(16*x - 3*x*exp(x
)) - 3*x^2*exp(x) + 16*x^2) - log(x + log(x))*log(log(x + log(x)))*(3*x*log(x) + 3*x^2)),x)

[Out]

log(log(log(x + log(x))) + exp(x) - 16/3)

________________________________________________________________________________________

sympy [A]  time = 2.51, size = 15, normalized size = 0.68 \begin {gather*} \log {\left (e^{x} + \log {\left (\log {\left (x + \log {\relax (x )} \right )} \right )} - \frac {16}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*ln(x)+3*exp(x)*x**2)*ln(x+ln(x))+3*x+3)/((3*x*ln(x)+3*x**2)*ln(x+ln(x))*ln(ln(x+ln(x)))
+((3*exp(x)*x-16*x)*ln(x)+3*exp(x)*x**2-16*x**2)*ln(x+ln(x))),x)

[Out]

log(exp(x) + log(log(x + log(x))) - 16/3)

________________________________________________________________________________________