3.14.45 \(\int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+(x-3 x^3+2 x^2 \log (4)) \log (x^2)+(4 x^2+2 x^2 \log (x^2)) \log (2+\log (x^2))}{2 x^2-2 x^4+2 x^3 \log (4)+(x^2-x^4+x^3 \log (4)) \log (x^2)+(-2 x+2 x^3+(-x+x^3) \log (x^2)) \log (2+\log (x^2))} \, dx\)

Optimal. Leaf size=30 \[ \log \left (x^2 \log (4)+\frac {\left (x-x^3\right ) \left (x-\log \left (2+\log \left (x^2\right )\right )\right )}{x}\right ) \]

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Rubi [A]  time = 0.49, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, integrand size = 134, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 6741, 6684} \begin {gather*} \log \left (-x^3+x^2 \log \left (\log \left (x^2\right )+2\right )+x^2 \log (4)-\log \left (\log \left (x^2\right )+2\right )+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 2*x + 2*x^2 - 6*x^3 + 4*x^2*Log[4] + (x - 3*x^3 + 2*x^2*Log[4])*Log[x^2] + (4*x^2 + 2*x^2*Log[x^2])*
Log[2 + Log[x^2]])/(2*x^2 - 2*x^4 + 2*x^3*Log[4] + (x^2 - x^4 + x^3*Log[4])*Log[x^2] + (-2*x + 2*x^3 + (-x + x
^3)*Log[x^2])*Log[2 + Log[x^2]]),x]

[Out]

Log[x - x^3 + x^2*Log[4] - Log[2 + Log[x^2]] + x^2*Log[2 + Log[x^2]]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx\\ &=\int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{x \left (2+\log \left (x^2\right )\right ) \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right )} \, dx\\ &=\log \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 34, normalized size = 1.13 \begin {gather*} \log \left (-x+x^3-x^2 \log (4)+\log \left (2+\log \left (x^2\right )\right )-x^2 \log \left (2+\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 2*x + 2*x^2 - 6*x^3 + 4*x^2*Log[4] + (x - 3*x^3 + 2*x^2*Log[4])*Log[x^2] + (4*x^2 + 2*x^2*Log[
x^2])*Log[2 + Log[x^2]])/(2*x^2 - 2*x^4 + 2*x^3*Log[4] + (x^2 - x^4 + x^3*Log[4])*Log[x^2] + (-2*x + 2*x^3 + (
-x + x^3)*Log[x^2])*Log[2 + Log[x^2]]),x]

[Out]

Log[-x + x^3 - x^2*Log[4] + Log[2 + Log[x^2]] - x^2*Log[2 + Log[x^2]]]

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fricas [A]  time = 1.16, size = 45, normalized size = 1.50 \begin {gather*} \log \left (x^{2} - 1\right ) + \log \left (-\frac {x^{3} - 2 \, x^{2} \log \relax (2) - {\left (x^{2} - 1\right )} \log \left (\log \left (x^{2}\right ) + 2\right ) - x}{x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*log(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x
-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log(2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x,
algorithm="fricas")

[Out]

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

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giac [A]  time = 0.50, size = 35, normalized size = 1.17 \begin {gather*} \log \left (-x^{3} + 2 \, x^{2} \log \relax (2) + x^{2} \log \left (\log \left (x^{2}\right ) + 2\right ) + x - \log \left (\log \left (x^{2}\right ) + 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*log(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x
-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log(2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x,
algorithm="giac")

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{2} \ln \left (x^{2}\right )+4 x^{2}\right ) \ln \left (2+\ln \left (x^{2}\right )\right )+\left (4 x^{2} \ln \relax (2)-3 x^{3}+x \right ) \ln \left (x^{2}\right )+8 x^{2} \ln \relax (2)-6 x^{3}+2 x^{2}+2 x -2}{\left (\left (x^{3}-x \right ) \ln \left (x^{2}\right )+2 x^{3}-2 x \right ) \ln \left (2+\ln \left (x^{2}\right )\right )+\left (2 x^{3} \ln \relax (2)-x^{4}+x^{2}\right ) \ln \left (x^{2}\right )+4 x^{3} \ln \relax (2)-2 x^{4}+2 x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*ln(x^2)+4*x^2)*ln(2+ln(x^2))+(4*x^2*ln(2)-3*x^3+x)*ln(x^2)+8*x^2*ln(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x
)*ln(x^2)+2*x^3-2*x)*ln(2+ln(x^2))+(2*x^3*ln(2)-x^4+x^2)*ln(x^2)+4*x^3*ln(2)-2*x^4+2*x^2),x)

[Out]

int(((2*x^2*ln(x^2)+4*x^2)*ln(2+ln(x^2))+(4*x^2*ln(2)-3*x^3+x)*ln(x^2)+8*x^2*ln(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x
)*ln(x^2)+2*x^3-2*x)*ln(2+ln(x^2))+(2*x^3*ln(2)-x^4+x^2)*ln(x^2)+4*x^3*ln(2)-2*x^4+2*x^2),x)

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maxima [A]  time = 0.96, size = 47, normalized size = 1.57 \begin {gather*} \log \left (x + 1\right ) + \log \left (x - 1\right ) + \log \left (-\frac {x^{3} - 3 \, x^{2} \log \relax (2) - {\left (x^{2} - 1\right )} \log \left (\log \relax (x) + 1\right ) - x + \log \relax (2)}{x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*log(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x
-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log(2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x,
algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x+\ln \left (x^2\right )\,\left (-3\,x^3+4\,\ln \relax (2)\,x^2+x\right )+\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x^2\,\ln \left (x^2\right )+4\,x^2\right )+8\,x^2\,\ln \relax (2)+2\,x^2-6\,x^3-2}{4\,x^3\,\ln \relax (2)-\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x+\ln \left (x^2\right )\,\left (x-x^3\right )-2\,x^3\right )+\ln \left (x^2\right )\,\left (-x^4+2\,\ln \relax (2)\,x^3+x^2\right )+2\,x^2-2\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + log(x^2)*(x + 4*x^2*log(2) - 3*x^3) + log(log(x^2) + 2)*(2*x^2*log(x^2) + 4*x^2) + 8*x^2*log(2) + 2
*x^2 - 6*x^3 - 2)/(4*x^3*log(2) - log(log(x^2) + 2)*(2*x + log(x^2)*(x - x^3) - 2*x^3) + log(x^2)*(2*x^3*log(2
) + x^2 - x^4) + 2*x^2 - 2*x^4),x)

[Out]

int((2*x + log(x^2)*(x + 4*x^2*log(2) - 3*x^3) + log(log(x^2) + 2)*(2*x^2*log(x^2) + 4*x^2) + 8*x^2*log(2) + 2
*x^2 - 6*x^3 - 2)/(4*x^3*log(2) - log(log(x^2) + 2)*(2*x + log(x^2)*(x - x^3) - 2*x^3) + log(x^2)*(2*x^3*log(2
) + x^2 - x^4) + 2*x^2 - 2*x^4), x)

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sympy [A]  time = 1.43, size = 34, normalized size = 1.13 \begin {gather*} \log {\left (x^{2} - 1 \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} + 2 \right )} + \frac {- x^{3} + 2 x^{2} \log {\relax (2 )} + x}{x^{2} - 1} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2*ln(x**2)+4*x**2)*ln(2+ln(x**2))+(4*x**2*ln(2)-3*x**3+x)*ln(x**2)+8*x**2*ln(2)-6*x**3+2*x**2
+2*x-2)/(((x**3-x)*ln(x**2)+2*x**3-2*x)*ln(2+ln(x**2))+(2*x**3*ln(2)-x**4+x**2)*ln(x**2)+4*x**3*ln(2)-2*x**4+2
*x**2),x)

[Out]

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

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