3.41.13 $\int \frac{-x^2+(-40x-2x^2)\log (20+x)\log (\log (20+x))}{(240+12x)\log (20+x)}dx$

Optimal. Leaf size=12 $$-\frac{1}{12}{x}^2\log (\log (20+x))$$

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Rubi [F]  time = 0.24, 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 = {} $$\begin{array}{c}\int \frac{-x^2+(-40x-2x^2)\log (20+x)\log (\log (20+x))}{(240+12x)\log (20+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-x^2 + (-40*x - 2*x^2)*Log[20 + x]*Log[Log[20 + x]])/((240 + 12*x)*Log[20 + x]),x]

[Out]

-1/12*ExpIntegralEi[2*Log[20 + x]] - (100*Log[Log[20 + x]])/3 + (10*LogIntegral[20 + x])/3 - Defer[Int][x*Log[
Log[20 + x]], x]/6

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{1}{12}x(-\frac{x}{(20+x)\log (20+x)}-2\log (\log (20+x)))dx\\ & =\frac{1}{12}\int x(-\frac{x}{(20+x)\log (20+x)}-2\log (\log (20+x)))dx\\ & =\frac{1}{12}\int (-\frac{x^2}{(20+x)\log (20+x)}-2x\log (\log (20+x)))dx\\ & =-(\frac{1}{12}\int \frac{x^2}{(20+x)\log (20+x)}dx)-\frac{1}{6}\int x\log (\log (20+x))dx\\ & =-\left(\frac{1}{12}\mathrm{Subst}(\int \frac{(-20+x{)}^2}{x\log (x)}dx,x,20+x)\right)-\frac{1}{6}\int x\log (\log (20+x))dx\\ & =-\left(\frac{1}{12}\mathrm{Subst}(\int (-\frac{40}{\log (x)}+\frac{400}{x\log (x)}+\frac{x}{\log (x)})dx,x,20+x)\right)-\frac{1}{6}\int x\log (\log (20+x))dx\\ & =-\left(\frac{1}{12}\mathrm{Subst}(\int \frac{x}{\log (x)}dx,x,20+x)\right)-\frac{1}{6}\int x\log (\log (20+x))dx+\frac{10}{3}\mathrm{Subst}(\int \frac{1}{\log (x)}dx,x,20+x)-\frac{100}{3}\mathrm{Subst}(\int \frac{1}{x\log (x)}dx,x,20+x)\\ & =\frac{10\text{li}(20+x)}{3}-\frac{1}{12}\mathrm{Subst}(\int \frac{e^{2x}}{x}dx,x,\log (20+x))-\frac{1}{6}\int x\log (\log (20+x))dx-\frac{100}{3}\mathrm{Subst}(\int \frac{1}{x}dx,x,\log (20+x))\\ & =-\frac{1}{12}\text{Ei}(2\log (20+x))-\frac{100}{3}\log (\log (20+x))+\frac{10\text{li}(20+x)}{3}-\frac{1}{6}\int x\log (\log (20+x))dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.06, size = 12, normalized size = 1.00 $$\begin{array}{c}-\frac{1}{12}{x}^2\log (\log (20+x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + (-40*x - 2*x^2)*Log[20 + x]*Log[Log[20 + x]])/((240 + 12*x)*Log[20 + x]),x]

[Out]

-1/12*(x^2*Log[Log[20 + x]])

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fricas [A]  time = 0.64, size = 10, normalized size = 0.83 $$\begin{array}{c}-\frac{1}{12}{x}^2\log (\log (x+20))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-40*x)*log(20+x)*log(log(20+x))-x^2)/(12*x+240)/log(20+x),x, algorithm="fricas")

[Out]

-1/12*x^2*log(log(x + 20))

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giac [A]  time = 0.27, size = 10, normalized size = 0.83 $$\begin{array}{c}-\frac{1}{12}{x}^2\log (\log (x+20))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-40*x)*log(20+x)*log(log(20+x))-x^2)/(12*x+240)/log(20+x),x, algorithm="giac")

[Out]

-1/12*x^2*log(log(x + 20))

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maple [A]  time = 0.09, size = 11, normalized size = 0.92

method	result	size
norman	$-\frac{x^2\ln (\ln (20+x))}{12}$	$11$
risch	$-\frac{x^2\ln (\ln (20+x))}{12}$	$11$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-40*x)*ln(20+x)*ln(ln(20+x))-x^2)/(12*x+240)/ln(20+x),x,method=_RETURNVERBOSE)

[Out]

-1/12*x^2*ln(ln(20+x))

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maxima [A]  time = 0.40, size = 10, normalized size = 0.83 $$\begin{array}{c}-\frac{1}{12}{x}^2\log (\log (x+20))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-40*x)*log(20+x)*log(log(20+x))-x^2)/(12*x+240)/log(20+x),x, algorithm="maxima")

[Out]

-1/12*x^2*log(log(x + 20))

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mupad [B]  time = 3.78, size = 10, normalized size = 0.83 $$\begin{array}{c}-\frac{x^2\ln (\ln (x+20))}{12}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 + log(x + 20)*log(log(x + 20))*(40*x + 2*x^2))/(log(x + 20)*(12*x + 240)),x)

[Out]

-(x^2*log(log(x + 20)))/12

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sympy [A]  time = 0.51, size = 24, normalized size = 2.00 $$\begin{array}{c}(\frac{100}{9}-\frac{x^2}{12})\log (\log (x+20))-\frac{100\log (\log (x+20))}{9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-40*x)*ln(20+x)*ln(ln(20+x))-x**2)/(12*x+240)/ln(20+x),x)

[Out]

(100/9 - x**2/12)*log(log(x + 20)) - 100*log(log(x + 20))/9

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