3.17.3 $\int \frac{-4x+(3+x)\log (3+x)\log ({\log }^4(3+x))}{(9+3x)\log (3+x){\log }^2({\log }^4(3+x))}dx$

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

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Rubi [F]  time = 0.45, 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 = {} $$\begin{array}{c}\int \frac{-4x+(3+x)\log (3+x)\log ({\log }^4(3+x))}{(9+3x)\log (3+x){\log }^2({\log }^4(3+x))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-4*x + (3 + x)*Log[3 + x]*Log[Log[3 + x]^4])/((9 + 3*x)*Log[3 + x]*Log[Log[3 + x]^4]^2),x]

[Out]

-Log[Log[3 + x]^4]^(-1) - (4*Defer[Subst][Defer[Int][1/(Log[x]*Log[Log[x]^4]^2), x], x, 3 + x])/3 + Defer[Subs
t][Defer[Int][Log[Log[x]^4]^(-1), x], x, 3 + x]/3

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 14, normalized size = 1.00 $$\begin{array}{c}\frac{x}{3\log ({\log }^4(3+x))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + (3 + x)*Log[3 + x]*Log[Log[3 + x]^4])/((9 + 3*x)*Log[3 + x]*Log[Log[3 + x]^4]^2),x]

[Out]

x/(3*Log[Log[3 + x]^4])

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fricas [A]  time = 0.62, size = 12, normalized size = 0.86 $$\begin{array}{c}\frac{x}{3\log (\log {(x+3)}^4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="fricas")

[Out]

1/3*x/log(log(x + 3)^4)

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giac [A]  time = 0.31, size = 12, normalized size = 0.86 $$\begin{array}{c}\frac{x}{3\log (\log {(x+3)}^4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="giac")

[Out]

1/3*x/log(log(x + 3)^4)

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maple [A]  time = 0.45, size = 13, normalized size = 0.93

method	result	size
norman	$\frac{x}{3\ln (\ln {(3+x)}^4)}$	$13$
risch	$\frac{2ix}{3(\pi \mathrm{csgn}{(i\ln (3+x))}^2\mathrm{csgn}(i\ln {(3+x)}^2)-2\pi \mathrm{csgn}(i\ln (3+x))\mathrm{csgn}{(i\ln {(3+x)}^2)}^2+\pi \mathrm{csgn}(i\ln (3+x))\mathrm{csgn}(i\ln {(3+x)}^2)\mathrm{csgn}(i\ln {(3+x)}^3)-\pi \mathrm{csgn}(i\ln (3+x))\mathrm{csgn}{(i\ln {(3+x)}^3)}^2+\pi \mathrm{csgn}(i\ln (3+x))\mathrm{csgn}(i\ln {(3+x)}^3)\mathrm{csgn}(i\ln {(3+x)}^4)-\pi \mathrm{csgn}(i\ln (3+x))\mathrm{csgn}{(i\ln {(3+x)}^4)}^2+\pi \mathrm{csgn}{(i\ln {(3+x)}^2)}^3-\pi \mathrm{csgn}(i\ln {(3+x)}^2)\mathrm{csgn}{(i\ln {(3+x)}^3)}^2+\pi \mathrm{csgn}{(i\ln {(3+x)}^3)}^3-\pi \mathrm{csgn}(i\ln {(3+x)}^3)\mathrm{csgn}{(i\ln {(3+x)}^4)}^2+\pi \mathrm{csgn}{(i\ln {(3+x)}^4)}^3+8i\ln (\ln (3+x)))}$	$259$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+x)*ln(3+x)*ln(ln(3+x)^4)-4*x)/(3*x+9)/ln(3+x)/ln(ln(3+x)^4)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 10, normalized size = 0.71 $$\begin{array}{c}\frac{x}{12\log (\log (x+3))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="maxima")

[Out]

1/12*x/log(log(x + 3))

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mupad [B]  time = 1.42, size = 12, normalized size = 0.86 $$\begin{array}{c}\frac{x}{3\ln \left({\ln (x+3)}^4\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - log(log(x + 3)^4)*log(x + 3)*(x + 3))/(log(log(x + 3)^4)^2*log(x + 3)*(3*x + 9)),x)

[Out]

x/(3*log(log(x + 3)^4))

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sympy [A]  time = 0.25, size = 10, normalized size = 0.71 $$\begin{array}{c}\frac{x}{3\log (\log {(x+3)}^4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*ln(3+x)*ln(ln(3+x)**4)-4*x)/(3*x+9)/ln(3+x)/ln(ln(3+x)**4)**2,x)

[Out]

x/(3*log(log(x + 3)**4))

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