3.40.81 \(\int \frac {1+x+2 e x^2+(-8 x+2 x^2+2 e x^3+2 x \log (x)) \log (4-x-e x^2-\log (x))}{(-4 x+x^2+e x^3+x \log (x)) \log (4-x-e x^2-\log (x))} \, dx\)

Optimal. Leaf size=22 \[ -3+2 x+\log \left (\log \left (4-x-e x^2-\log (x)\right )\right ) \]

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Rubi [A]  time = 0.34, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6742, 6684} \begin {gather*} \log \left (\log \left (-e x^2-x-\log (x)+4\right )\right )+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x + 2*E*x^2 + (-8*x + 2*x^2 + 2*E*x^3 + 2*x*Log[x])*Log[4 - x - E*x^2 - Log[x]])/((-4*x + x^2 + E*x^3
+ x*Log[x])*Log[4 - x - E*x^2 - Log[x]]),x]

[Out]

2*x + Log[Log[4 - x - E*x^2 - Log[x]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {1+x+2 e x^2}{x \left (-4+x+e x^2+\log (x)\right ) \log \left (4-x-e x^2-\log (x)\right )}\right ) \, dx\\ &=2 x+\int \frac {1+x+2 e x^2}{x \left (-4+x+e x^2+\log (x)\right ) \log \left (4-x-e x^2-\log (x)\right )} \, dx\\ &=2 x+\log \left (\log \left (4-x-e x^2-\log (x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 21, normalized size = 0.95 \begin {gather*} 2 x+\log \left (\log \left (4-x-e x^2-\log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + 2*E*x^2 + (-8*x + 2*x^2 + 2*E*x^3 + 2*x*Log[x])*Log[4 - x - E*x^2 - Log[x]])/((-4*x + x^2 +
E*x^3 + x*Log[x])*Log[4 - x - E*x^2 - Log[x]]),x]

[Out]

2*x + Log[Log[4 - x - E*x^2 - Log[x]]]

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fricas [A]  time = 0.83, size = 22, normalized size = 1.00 \begin {gather*} 2 \, x + \log \left (\log \left (-x^{2} e - x - \log \relax (x) + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^3*exp(1)+2*x^2-8*x)*log(-log(x)-x^2*exp(1)-x+4)+2*x^2*exp(1)+x+1)/(x*log(x)+x^3*exp
(1)+x^2-4*x)/log(-log(x)-x^2*exp(1)-x+4),x, algorithm="fricas")

[Out]

2*x + log(log(-x^2*e - x - log(x) + 4))

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giac [A]  time = 0.20, size = 22, normalized size = 1.00 \begin {gather*} 2 \, x + \log \left (\log \left (-x^{2} e - x - \log \relax (x) + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^3*exp(1)+2*x^2-8*x)*log(-log(x)-x^2*exp(1)-x+4)+2*x^2*exp(1)+x+1)/(x*log(x)+x^3*exp
(1)+x^2-4*x)/log(-log(x)-x^2*exp(1)-x+4),x, algorithm="giac")

[Out]

2*x + log(log(-x^2*e - x - log(x) + 4))

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maple [A]  time = 0.04, size = 23, normalized size = 1.05

method	result	size
risch	\(2 x +\ln \left (\ln \left (-\ln \relax (x )-x^{2} {\mathrm e}-x +4\right )\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)+2*x^3*exp(1)+2*x^2-8*x)*ln(-ln(x)-x^2*exp(1)-x+4)+2*x^2*exp(1)+x+1)/(x*ln(x)+x^3*exp(1)+x^2-4*
x)/ln(-ln(x)-x^2*exp(1)-x+4),x,method=_RETURNVERBOSE)

[Out]

2*x+ln(ln(-ln(x)-x^2*exp(1)-x+4))

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maxima [A]  time = 0.46, size = 22, normalized size = 1.00 \begin {gather*} 2 \, x + \log \left (\log \left (-x^{2} e - x - \log \relax (x) + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x^3*exp(1)+2*x^2-8*x)*log(-log(x)-x^2*exp(1)-x+4)+2*x^2*exp(1)+x+1)/(x*log(x)+x^3*exp
(1)+x^2-4*x)/log(-log(x)-x^2*exp(1)-x+4),x, algorithm="maxima")

[Out]

2*x + log(log(-x^2*e - x - log(x) + 4))

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mupad [B]  time = 3.03, size = 22, normalized size = 1.00 \begin {gather*} 2\,x+\ln \left (\ln \left (4-\ln \relax (x)-x^2\,\mathrm {e}-x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*x^2*exp(1) + log(4 - log(x) - x^2*exp(1) - x)*(2*x^3*exp(1) - 8*x + 2*x*log(x) + 2*x^2) + 1)/(log(4
- log(x) - x^2*exp(1) - x)*(x^3*exp(1) - 4*x + x*log(x) + x^2)),x)

[Out]

2*x + log(log(4 - log(x) - x^2*exp(1) - x))

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sympy [A]  time = 0.51, size = 19, normalized size = 0.86 \begin {gather*} 2 x + \log {\left (\log {\left (- e x^{2} - x - \log {\relax (x )} + 4 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)+2*x**3*exp(1)+2*x**2-8*x)*ln(-ln(x)-x**2*exp(1)-x+4)+2*x**2*exp(1)+x+1)/(x*ln(x)+x**3*ex
p(1)+x**2-4*x)/ln(-ln(x)-x**2*exp(1)-x+4),x)

[Out]

2*x + log(log(-E*x**2 - x - log(x) + 4))

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