\(\int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx\) [3253]

Optimal result

Integrand size = 49, antiderivative size = 14 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=\log \left (\frac {16}{\left (1+x+(3+\log (x))^x\right )^4}\right ) \]

[Out]

ln(16/(x+exp(x*ln(3+ln(x)))+1)^4)

Rubi [F]

\[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=\int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx \]

[In]

Int[(-12 - 4*Log[x] + (3 + Log[x])^x*(-4 + (-12 - 4*Log[x])*Log[3 + Log[x]]))/(3 + 3*x + (1 + x)*Log[x] + (3 +
Log[x])^(1 + x)),x]

[Out]

(-4*ExpIntegralEi[3 + Log[x]])/E^3 - 8*Defer[Int][1/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int]
[x/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] - 4*Defer[Int][Log[x]/((3 + Log[x])*(1 + x + (3 + Log[x])^x)),
x] - 4*Defer[Int][Log[3 + Log[x]], x] + 12*Defer[Int][Log[3 + Log[x]]/((3 + Log[x])*(1 + x + (3 + Log[x])^x)),
x] + 12*Defer[Int][(x*Log[3 + Log[x]])/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int][(Log[x]*Log
[3 + Log[x]])/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int][(x*Log[x]*Log[3 + Log[x]])/((3 + Log[
x])*(1 + x + (3 + Log[x])^x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx \\ & = \int \left (-\frac {4 (1+3 \log (3+\log (x))+\log (x) \log (3+\log (x)))}{3+\log (x)}+\frac {4 (-2+x-\log (x)+3 \log (3+\log (x))+3 x \log (3+\log (x))+\log (x) \log (3+\log (x))+x \log (x) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}\right ) \, dx \\ & = -\left (4 \int \frac {1+3 \log (3+\log (x))+\log (x) \log (3+\log (x))}{3+\log (x)} \, dx\right )+4 \int \frac {-2+x-\log (x)+3 \log (3+\log (x))+3 x \log (3+\log (x))+\log (x) \log (3+\log (x))+x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx \\ & = -\left (4 \int \left (\frac {1}{3+\log (x)}+\log (3+\log (x))\right ) \, dx\right )+4 \int \frac {-2+x+3 (1+x) \log (3+\log (x))+\log (x) (-1+(1+x) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx \\ & = -\left (4 \int \frac {1}{3+\log (x)} \, dx\right )-4 \int \log (3+\log (x)) \, dx+4 \int \left (-\frac {2}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}-\frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {3 \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {3 x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}\right ) \, dx \\ & = 4 \int \frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \log (3+\log (x)) \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+4 \int \frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \text {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (x)\right )-8 \int \frac {1}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {\log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx \\ & = -\frac {4 \text {Ei}(3+\log (x))}{e^3}+4 \int \frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \log (3+\log (x)) \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+4 \int \frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-8 \int \frac {1}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {\log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=-4 \log \left (1+x+(3+\log (x))^x\right ) \]

[In]

Integrate[(-12 - 4*Log[x] + (3 + Log[x])^x*(-4 + (-12 - 4*Log[x])*Log[3 + Log[x]]))/(3 + 3*x + (1 + x)*Log[x]
+ (3 + Log[x])^(1 + x)),x]

[Out]

-4*Log[1 + x + (3 + Log[x])^x]

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
risch	\(-4 \ln \left (\left (3+\ln \left (x \right )\right )^{x}+x +1\right )\)	\(13\)
parallelrisch	\(-4 \ln \left (x +{\mathrm e}^{x \ln \left (3+\ln \left (x \right )\right )}+1\right )\)	\(15\)

[In]

int((((-4*ln(x)-12)*ln(3+ln(x))-4)*exp(x*ln(3+ln(x)))-4*ln(x)-12)/((3+ln(x))*exp(x*ln(3+ln(x)))+ln(x)*(1+x)+3*
x+3),x,method=_RETURNVERBOSE)

[Out]

-4*ln((3+ln(x))^x+x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=-4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \]

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="fricas")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=- 4 \log {\left (x + e^{x \log {\left (\log {\left (x \right )} + 3 \right )}} + 1 \right )} \]

[In]

integrate((((-4*ln(x)-12)*ln(3+ln(x))-4)*exp(x*ln(3+ln(x)))-4*ln(x)-12)/((3+ln(x))*exp(x*ln(3+ln(x)))+ln(x)*(1
+x)+3*x+3),x)

[Out]

-4*log(x + exp(x*log(log(x) + 3)) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=-4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \]

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="maxima")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (14) = 28\).

Time = 0.90 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.79 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=-\frac {4 \, x e^{3} \log \left (x\right ) \log \left (\log \left (x\right ) + 3\right )}{e^{3} \log \left (x\right ) + 3 \, e^{3}} + 4 \, x \log \left (\log \left (x\right ) + 3\right ) - \frac {12 \, x e^{3} \log \left (\log \left (x\right ) + 3\right )}{e^{3} \log \left (x\right ) + 3 \, e^{3}} - 4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \]

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="giac")

[Out]

-4*x*e^3*log(x)*log(log(x) + 3)/(e^3*log(x) + 3*e^3) + 4*x*log(log(x) + 3) - 12*x*e^3*log(log(x) + 3)/(e^3*log
(x) + 3*e^3) - 4*log(x + (log(x) + 3)^x + 1)

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx=-4\,\ln \left (x+{\left (\ln \left (x\right )+3\right )}^x+1\right ) \]

[In]

int(-(4*log(x) + exp(x*log(log(x) + 3))*(log(log(x) + 3)*(4*log(x) + 12) + 4) + 12)/(3*x + log(x)*(x + 1) + ex
p(x*log(log(x) + 3))*(log(x) + 3) + 3),x)

[Out]

-4*log(x + (log(x) + 3)^x + 1)