3.85.32 $\int \frac{6+32x^2+(-3+581x+194x^2+3072x^3+1024x^4)\log (3+x)+(2+(192x+64x^2)\log (3+x))\log (\log (3+x))}{(3+x)\log (3+x)}dx$

Optimal. Leaf size=23 $$-x+x^2+{(-3-16x^2-\log (\log (3+x)))}^2$$

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Rubi [F]  time = 0.39, 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 = {} $$\begin{array}{c}\int \frac{6+32x^2+(-3+581x+194x^2+3072x^3+1024x^4)\log (3+x)+(2+(192x+64x^2)\log (3+x))\log (\log (3+x))}{(3+x)\log (3+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(6 + 32*x^2 + (-3 + 581*x + 194*x^2 + 3072*x^3 + 1024*x^4)*Log[3 + x] + (2 + (192*x + 64*x^2)*Log[3 + x])*
Log[Log[3 + x]])/((3 + x)*Log[3 + x]),x]

[Out]

-x + 97*x^2 + 256*x^4 + 32*ExpIntegralEi[2*Log[3 + x]] + 294*Log[Log[3 + x]] + Log[Log[3 + x]]^2 - 192*LogInte
gral[3 + x] + 64*Defer[Int][x*Log[Log[3 + x]], x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-1+194x+1024x^3+64x\log (\log (3+x))+\frac{2(3+16x^2+\log (\log (3+x)))}{(3+x)\log (3+x)})dx\\ & =-x+97x^2+256x^4+2\int \frac{3+16x^2+\log (\log (3+x))}{(3+x)\log (3+x)}dx+64\int x\log (\log (3+x))dx\\ & =-x+97x^2+256x^4+2\int (\frac{3+16x^2}{(3+x)\log (3+x)}+\frac{\log (\log (3+x))}{(3+x)\log (3+x)})dx+64\int x\log (\log (3+x))dx\\ & =-x+97x^2+256x^4+2\int \frac{3+16x^2}{(3+x)\log (3+x)}dx+2\int \frac{\log (\log (3+x))}{(3+x)\log (3+x)}dx+64\int x\log (\log (3+x))dx\\ & =-x+97x^2+256x^4+{\log }^2(\log (3+x))+2\int (-\frac{48}{\log (3+x)}+\frac{16x}{\log (3+x)}+\frac{147}{(3+x)\log (3+x)})dx+64\int x\log (\log (3+x))dx\\ & =-x+97x^2+256x^4+{\log }^2(\log (3+x))+32\int \frac{x}{\log (3+x)}dx+64\int x\log (\log (3+x))dx-96\int \frac{1}{\log (3+x)}dx+294\int \frac{1}{(3+x)\log (3+x)}dx\\ & =-x+97x^2+256x^4+{\log }^2(\log (3+x))+32\int (-\frac{3}{\log (3+x)}+\frac{3+x}{\log (3+x)})dx+64\int x\log (\log (3+x))dx-96\mathrm{Subst}(\int \frac{1}{\log (x)}dx,x,3+x)+294\mathrm{Subst}(\int \frac{1}{x\log (x)}dx,x,3+x)\\ & =-x+97x^2+256x^4+{\log }^2(\log (3+x))-96\text{li}(3+x)+32\int \frac{3+x}{\log (3+x)}dx+64\int x\log (\log (3+x))dx-96\int \frac{1}{\log (3+x)}dx+294\mathrm{Subst}(\int \frac{1}{x}dx,x,\log (3+x))\\ & =-x+97x^2+256x^4+294\log (\log (3+x))+{\log }^2(\log (3+x))-96\text{li}(3+x)+32\mathrm{Subst}(\int \frac{x}{\log (x)}dx,x,3+x)+64\int x\log (\log (3+x))dx-96\mathrm{Subst}(\int \frac{1}{\log (x)}dx,x,3+x)\\ & =-x+97x^2+256x^4+294\log (\log (3+x))+{\log }^2(\log (3+x))-192\text{li}(3+x)+32\mathrm{Subst}(\int \frac{e^{2x}}{x}dx,x,\log (3+x))+64\int x\log (\log (3+x))dx\\ & =-x+97x^2+256x^4+32\text{Ei}(2\log (3+x))+294\log (\log (3+x))+{\log }^2(\log (3+x))-192\text{li}(3+x)+64\int x\log (\log (3+x))dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.10, size = 41, normalized size = 1.78 $$\begin{array}{c}-x+97x^2+256x^4+294\log (\log (3+x))+32(-3+x)(3+x)\log (\log (3+x))+{\log }^2(\log (3+x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(6 + 32*x^2 + (-3 + 581*x + 194*x^2 + 3072*x^3 + 1024*x^4)*Log[3 + x] + (2 + (192*x + 64*x^2)*Log[3
+ x])*Log[Log[3 + x]])/((3 + x)*Log[3 + x]),x]

[Out]

-x + 97*x^2 + 256*x^4 + 294*Log[Log[3 + x]] + 32*(-3 + x)*(3 + x)*Log[Log[3 + x]] + Log[Log[3 + x]]^2

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fricas [A]  time = 1.12, size = 35, normalized size = 1.52 $$\begin{array}{c}256x^4+97x^2+2(16x^2+3)\log (\log (x+3))+\log {(\log (x+3))}^2-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="fricas")

[Out]

256*x^4 + 97*x^2 + 2*(16*x^2 + 3)*log(log(x + 3)) + log(log(x + 3))^2 - x

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giac [A]  time = 0.23, size = 38, normalized size = 1.65 $$\begin{array}{c}256x^4+32x^2\log (\log (x+3))+97x^2+\log {(\log (x+3))}^2-x+6\log (\log (x+3))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="giac")

[Out]

256*x^4 + 32*x^2*log(log(x + 3)) + 97*x^2 + log(log(x + 3))^2 - x + 6*log(log(x + 3))

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maple [A]  time = 0.37, size = 39, normalized size = 1.70

method	result	size
risch	$\ln {(\ln (3+x))}^2+32x^2\ln (\ln (3+x))+256x^4+97x^2-x+6\ln (\ln (3+x))$	$39$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((64*x^2+192*x)*ln(3+x)+2)*ln(ln(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*ln(3+x)+32*x^2+6)/(3+x)/ln(3+x
),x,method=_RETURNVERBOSE)

[Out]

ln(ln(3+x))^2+32*x^2*ln(ln(3+x))+256*x^4+97*x^2-x+6*ln(ln(3+x))

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maxima [A]  time = 0.40, size = 38, normalized size = 1.65 $$\begin{array}{c}256x^4+32x^2\log (\log (x+3))+97x^2+\log {(\log (x+3))}^2-x+6\log (\log (x+3))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="maxima")

[Out]

256*x^4 + 32*x^2*log(log(x + 3)) + 97*x^2 + log(log(x + 3))^2 - x + 6*log(log(x + 3))

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mupad [B]  time = 5.30, size = 38, normalized size = 1.65 $$\begin{array}{c}256x^4+32x^2\ln (\ln (x+3))+97x^2-x+{\ln (\ln (x+3))}^2+6\ln (\ln (x+3))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)*(581*x + 194*x^2 + 3072*x^3 + 1024*x^4 - 3) + log(log(x + 3))*(log(x + 3)*(192*x + 64*x^2) + 2
) + 32*x^2 + 6)/(log(x + 3)*(x + 3)),x)

[Out]

6*log(log(x + 3)) - x + log(log(x + 3))^2 + 97*x^2 + 256*x^4 + 32*x^2*log(log(x + 3))

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sympy [A]  time = 0.39, size = 39, normalized size = 1.70 $$\begin{array}{c}256x^4+32x^2\log (\log (x+3))+97x^2-x+\log {(\log (x+3))}^2+6\log (\log (x+3))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x**2+192*x)*ln(3+x)+2)*ln(ln(3+x))+(1024*x**4+3072*x**3+194*x**2+581*x-3)*ln(3+x)+32*x**2+6)/(
3+x)/ln(3+x),x)

[Out]

256*x**4 + 32*x**2*log(log(x + 3)) + 97*x**2 - x + log(log(x + 3))**2 + 6*log(log(x + 3))

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