3.26.95 \(\int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+(2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)) \log (-1+x+4 x^2+4 x^3+x^4+x \log (4 x))}{1-x-4 x^2-4 x^3-x^4-x \log (4 x)+(-1+x+4 x^2+4 x^3+x^4+x \log (4 x)) \log (-1+x+4 x^2+4 x^3+x^4+x \log (4 x))} \, dx\)

Optimal. Leaf size=30 \[ \log \left (e^{-2 x} \left (-1+\log \left (-1+x+x^2 (2+x)^2+x \log (4 x)\right )\right )^2\right ) \]

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Rubi [A]  time = 0.81, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, integrand size = 157, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6741, 6742, 6684} \begin {gather*} 2 \log \left (1-\log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )\right )-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 18*x + 32*x^2 + 16*x^3 + 2*x^4 + (2 + 2*x)*Log[4*x] + (2 - 2*x - 8*x^2 - 8*x^3 - 2*x^4 - 2*x*Log[4*x]
)*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]])/(1 - x - 4*x^2 - 4*x^3 - x^4 - x*Log[4*x] + (-1 + x + 4*x^2
+ 4*x^3 + x^4 + x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]]),x]

[Out]

-2*x + 2*Log[1 - Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*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 6741

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

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 \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{\left (1-x-4 x^2-4 x^3-x^4-x \log (4 x)\right ) \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )} \, dx\\ &=\int \left (-2+\frac {2 \left (2+8 x+12 x^2+4 x^3+\log (4 x)\right )}{\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \left (-1+\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )}\right ) \, dx\\ &=-2 x+2 \int \frac {2+8 x+12 x^2+4 x^3+\log (4 x)}{\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \left (-1+\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )} \, dx\\ &=-2 x+2 \log \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 34, normalized size = 1.13 \begin {gather*} 2 \left (-x+\log \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 18*x + 32*x^2 + 16*x^3 + 2*x^4 + (2 + 2*x)*Log[4*x] + (2 - 2*x - 8*x^2 - 8*x^3 - 2*x^4 - 2*x*Lo
g[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]])/(1 - x - 4*x^2 - 4*x^3 - x^4 - x*Log[4*x] + (-1 + x +
4*x^2 + 4*x^3 + x^4 + x*Log[4*x])*Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]]),x]

[Out]

2*(-x + Log[1 - Log[-1 + x + 4*x^2 + 4*x^3 + x^4 + x*Log[4*x]]])

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fricas [A]  time = 0.61, size = 32, normalized size = 1.07 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)+(2*x+2)*log(4*x)+2*x^4+
16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4+4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x
^3-4*x^2-x+1),x, algorithm="fricas")

[Out]

-2*x + 2*log(log(x^4 + 4*x^3 + 4*x^2 + x*log(4*x) + x - 1) - 1)

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giac [A]  time = 0.56, size = 32, normalized size = 1.07 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)+(2*x+2)*log(4*x)+2*x^4+
16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4+4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x
^3-4*x^2-x+1),x, algorithm="giac")

[Out]

-2*x + 2*log(log(x^4 + 4*x^3 + 4*x^2 + x*log(4*x) + x - 1) - 1)

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maple [A]  time = 0.03, size = 33, normalized size = 1.10

method	result	size
risch	\(-2 x +2 \ln \left (\ln \left (x \ln \left (4 x \right )+x^{4}+4 x^{3}+4 x^{2}+x -1\right )-1\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*ln(x*ln(4*x)+x^4+4*x^3+4*x^2+x-1)+(2*x+2)*ln(4*x)+2*x^4+16*x^3+32*
x^2+18*x+2)/((x*ln(4*x)+x^4+4*x^3+4*x^2+x-1)*ln(x*ln(4*x)+x^4+4*x^3+4*x^2+x-1)-x*ln(4*x)-x^4-4*x^3-4*x^2-x+1),
x,method=_RETURNVERBOSE)

[Out]

-2*x+2*ln(ln(x*ln(4*x)+x^4+4*x^3+4*x^2+x-1)-1)

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maxima [A]  time = 1.05, size = 37, normalized size = 1.23 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x {\left (2 \, \log \relax (2) + 1\right )} + x \log \relax (x) - 1\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(4*x)-2*x^4-8*x^3-8*x^2-2*x+2)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)+(2*x+2)*log(4*x)+2*x^4+
16*x^3+32*x^2+18*x+2)/((x*log(4*x)+x^4+4*x^3+4*x^2+x-1)*log(x*log(4*x)+x^4+4*x^3+4*x^2+x-1)-x*log(4*x)-x^4-4*x
^3-4*x^2-x+1),x, algorithm="maxima")

[Out]

-2*x + 2*log(log(x^4 + 4*x^3 + 4*x^2 + x*(2*log(2) + 1) + x*log(x) - 1) - 1)

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mupad [B]  time = 1.82, size = 32, normalized size = 1.07 \begin {gather*} 2\,\ln \left (\ln \left (x+x\,\ln \left (4\,x\right )+4\,x^2+4\,x^3+x^4-1\right )-1\right )-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x + 32*x^2 + 16*x^3 + 2*x^4 + log(4*x)*(2*x + 2) - log(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 - 1)*(2*x
+ 2*x*log(4*x) + 8*x^2 + 8*x^3 + 2*x^4 - 2) + 2)/(x + x*log(4*x) - log(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 -
1)*(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 - 1) + 4*x^2 + 4*x^3 + x^4 - 1),x)

[Out]

2*log(log(x + x*log(4*x) + 4*x^2 + 4*x^3 + x^4 - 1) - 1) - 2*x

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sympy [A]  time = 0.73, size = 32, normalized size = 1.07 \begin {gather*} - 2 x + 2 \log {\left (\log {\left (x^{4} + 4 x^{3} + 4 x^{2} + x \log {\left (4 x \right )} + x - 1 \right )} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(4*x)-2*x**4-8*x**3-8*x**2-2*x+2)*ln(x*ln(4*x)+x**4+4*x**3+4*x**2+x-1)+(2*x+2)*ln(4*x)+2*x*
*4+16*x**3+32*x**2+18*x+2)/((x*ln(4*x)+x**4+4*x**3+4*x**2+x-1)*ln(x*ln(4*x)+x**4+4*x**3+4*x**2+x-1)-x*ln(4*x)-
x**4-4*x**3-4*x**2-x+1),x)

[Out]

-2*x + 2*log(log(x**4 + 4*x**3 + 4*x**2 + x*log(4*x) + x - 1) - 1)

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