3.33.53 $\int \frac{-12-4\log (x)+(3+\log (x){)}^x(-4+(-12-4\log (x))\log (3+\log (x)))}{3+3x+(1+x)\log (x)+(3+\log (x){)}^{1+x}}dx$

Optimal. Leaf size=14 $$\log \left(\frac{16}{{(1+x+(3+\log (x){)}^x)}^4}\right)$$

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Rubi [F]  time = 2.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{-12-4\log (x)+(3+\log (x){)}^x(-4+(-12-4\log (x))\log (3+\log (x)))}{3+3x+(1+x)\log (x)+(3+\log (x){)}^{1+x}}dx\end{array}$$

Verification is not applicable to the result.

[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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-12-4\log (x)+(3+\log (x){)}^x(-4+(-12-4\log (x))\log (3+\log (x)))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx\\ & =\int (-\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))+3x\log (3+\log (x))+\log (x)\log (3+\log (x))+x\log (x)\log (3+\log (x)))}{(3+\log (x))(1+x+(3+\log (x){)}^x)})dx\\ & =-(4\int \frac{1+3\log (3+\log (x))+\log (x)\log (3+\log (x))}{3+\log (x)}dx)+4\int \frac{-2+x-\log (x)+3\log (3+\log (x))+3x\log (3+\log (x))+\log (x)\log (3+\log (x))+x\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx\\ & =-(4\int (\frac{1}{3+\log (x)}+\log (3+\log (x)))dx)+4\int \frac{-2+x+3(1+x)\log (3+\log (x))+\log (x)(-1+(1+x)\log (3+\log (x)))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx\\ & =-(4\int \frac{1}{3+\log (x)}dx)-4\int \log (3+\log (x))dx+4\int (-\frac{2}{(3+\log (x))(1+x+(3+\log (x){)}^x)}+\frac{x}{(3+\log (x))(1+x+(3+\log (x){)}^x)}-\frac{\log (x)}{(3+\log (x))(1+x+(3+\log (x){)}^x)}+\frac{3\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}+\frac{3x\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}+\frac{\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}+\frac{x\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)})dx\\ & =4\int \frac{x}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-4\int \frac{\log (x)}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-4\int \log (3+\log (x))dx+4\int \frac{\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+4\int \frac{x\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-4\mathrm{Subst}(\int \frac{e^x}{3+x}dx,x,\log (x))-8\int \frac{1}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+12\int \frac{\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+12\int \frac{x\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx\\ & =-\frac{4\text{Ei}(3+\log (x))}{e^3}+4\int \frac{x}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-4\int \frac{\log (x)}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-4\int \log (3+\log (x))dx+4\int \frac{\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+4\int \frac{x\log (x)\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx-8\int \frac{1}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+12\int \frac{\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx+12\int \frac{x\log (3+\log (x))}{(3+\log (x))(1+x+(3+\log (x){)}^x)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.37, size = 12, normalized size = 0.86 $$\begin{array}{c}-4\log (1+x+(3+\log (x){)}^x)\end{array}$$

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.67, size = 12, normalized size = 0.86 $$\begin{array}{c}-4\log (x+{(\log (x)+3)}^x+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)*(x+1)+3*x+3),x, algorithm="fricas")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

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giac [B]  time = 1.45, size = 67, normalized size = 4.79 $$\begin{array}{c}-\frac{4x{e}^3\log (x)\log (\log (x)+3)}{e^3\log (x)+3e^3}+4x\log (\log (x)+3)-\frac{12x{e}^3\log (\log (x)+3)}{e^3\log (x)+3e^3}-4\log (x+{(\log (x)+3)}^x+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)*(x+1)+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)

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maple [A]  time = 0.07, size = 13, normalized size = 0.93

method	result	size
risch	$-4\ln ({(3+\ln (x))}^x+x+1)$	$13$

Verification of antiderivative is not currently implemented for this CAS.

[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)*(x+1)+3*
x+3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.46, size = 12, normalized size = 0.86 $$\begin{array}{c}-4\log (x+{(\log (x)+3)}^x+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)*(x+1)+3*x+3),x, algorithm="maxima")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

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mupad [B]  time = 2.15, size = 12, normalized size = 0.86 $$\begin{array}{c}-4\ln (x+{(\ln (x)+3)}^x+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.84, size = 17, normalized size = 1.21 $$\begin{array}{c}-4\log (x+e^{x\log (\log (x)+3)}+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)*(x
+1)+3*x+3),x)

[Out]

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

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