3.77.59 \(\int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+(2 x^3+4 x^5+(-2 x^2-6 x^4) \log (4)+2 x^3 \log ^2(4)) \log (\frac {3}{x})+(4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)) \log (\frac {3}{x}) \log (\log (\frac {3}{x}))+(-2 \log (3)+(2 x+4 x^3-2 x^2 \log (4)) \log (\frac {3}{x})+(4 x^2 \log (3)-2 x \log (3) \log (4)) \log (\frac {3}{x}) \log (\log (\frac {3}{x}))) \log (\frac {x+\log (3) \log (\log (\frac {3}{x}))}{\log (3)})}{x^2 \log (\frac {3}{x})+x \log (3) \log (\frac {3}{x}) \log (\log (\frac {3}{x}))} \, dx\)

Optimal. Leaf size=26 \[ \left (x^2-x \log (4)+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \]

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Rubi [A]  time = 0.42, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 205, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6688, 12, 6686} \begin {gather*} \left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4)*Log[4] + 2*x^3*Log[4]^2)*Log[3/x] +
(4*x^4*Log[3] - 6*x^3*Log[3]*Log[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x + 4*x
^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3/x]*Log[Log[3/x]])*Log[(x + Log[3]*Log[L
og[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log[3]*Log[3/x]*Log[Log[3/x]]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (1+2 x^2-x \log (4)+(-\log (3) \log (4)+x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )\right )\right ) \left (-x (x-\log (4))-\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=2 \int \frac {\left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (1+2 x^2-x \log (4)+(-\log (3) \log (4)+x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )\right )\right ) \left (-x (x-\log (4))-\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4)*Log[4] + 2*x^3*Log[4]^2)*Log[
3/x] + (4*x^4*Log[3] - 6*x^3*Log[3]*Log[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x
+ 4*x^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3/x]*Log[Log[3/x]])*Log[(x + Log[3]
*Log[Log[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log[3]*Log[3/x]*Log[Log[3/x]]),x]

[Out]

Integrate[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4)*Log[4] + 2*x^3*Log[4]^2)*Log[
3/x] + (4*x^4*Log[3] - 6*x^3*Log[3]*Log[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x
+ 4*x^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3/x]*Log[Log[3/x]])*Log[(x + Log[3]
*Log[Log[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log[3]*Log[3/x]*Log[Log[3/x]]), x]

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fricas [B]  time = 1.49, size = 69, normalized size = 2.65 \begin {gather*} x^{4} - 4 \, x^{3} \log \relax (2) + 4 \, x^{2} \log \relax (2)^{2} + 2 \, {\left (x^{2} - 2 \, x \log \relax (2)\right )} \log \left (\frac {\log \relax (3) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \relax (3)}\right ) + \log \left (\frac {\log \relax (3) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \relax (3)}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(
3))*log((log(3)*log(log(3/x))+x)/log(3))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*lo
g(log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4*x*log(2)*log(3)-2*x^2*log(3))/(x*l
og(3)*log(3/x)*log(log(3/x))+x^2*log(3/x)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (x^{2} \log \relax (3) - 2 \, x \log \relax (3) \log \relax (2) - 2 \, {\left (x^{4} \log \relax (3) - 3 \, x^{3} \log \relax (3) \log \relax (2) + 2 \, x^{2} \log \relax (3) \log \relax (2)^{2}\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - {\left (2 \, {\left (x^{2} \log \relax (3) - x \log \relax (3) \log \relax (2)\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + {\left (2 \, x^{3} - 2 \, x^{2} \log \relax (2) + x\right )} \log \left (\frac {3}{x}\right ) - \log \relax (3)\right )} \log \left (\frac {\log \relax (3) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \relax (3)}\right ) - {\left (2 \, x^{5} + 4 \, x^{3} \log \relax (2)^{2} + x^{3} - 2 \, {\left (3 \, x^{4} + x^{2}\right )} \log \relax (2)\right )} \log \left (\frac {3}{x}\right )\right )}}{x \log \relax (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x^{2} \log \left (\frac {3}{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(
3))*log((log(3)*log(log(3/x))+x)/log(3))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*lo
g(log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4*x*log(2)*log(3)-2*x^2*log(3))/(x*l
og(3)*log(3/x)*log(log(3/x))+x^2*log(3/x)),x, algorithm="giac")

[Out]

integrate(-2*(x^2*log(3) - 2*x*log(3)*log(2) - 2*(x^4*log(3) - 3*x^3*log(3)*log(2) + 2*x^2*log(3)*log(2)^2)*lo
g(3/x)*log(log(3/x)) - (2*(x^2*log(3) - x*log(3)*log(2))*log(3/x)*log(log(3/x)) + (2*x^3 - 2*x^2*log(2) + x)*l
og(3/x) - log(3))*log((log(3)*log(log(3/x)) + x)/log(3)) - (2*x^5 + 4*x^3*log(2)^2 + x^3 - 2*(3*x^4 + x^2)*log
(2))*log(3/x))/(x*log(3)*log(3/x)*log(log(3/x)) + x^2*log(3/x)), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-4 x \ln \relax (2) \ln \relax (3)+4 x^{2} \ln \relax (3)\right ) \ln \left (\frac {3}{x}\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+\left (-4 x^{2} \ln \relax (2)+4 x^{3}+2 x \right ) \ln \left (\frac {3}{x}\right )-2 \ln \relax (3)\right ) \ln \left (\frac {\ln \relax (3) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \relax (3)}\right )+\left (8 x^{2} \ln \relax (3) \ln \relax (2)^{2}-12 x^{3} \ln \relax (3) \ln \relax (2)+4 x^{4} \ln \relax (3)\right ) \ln \left (\frac {3}{x}\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+\left (8 x^{3} \ln \relax (2)^{2}+2 \left (-6 x^{4}-2 x^{2}\right ) \ln \relax (2)+4 x^{5}+2 x^{3}\right ) \ln \left (\frac {3}{x}\right )+4 x \ln \relax (2) \ln \relax (3)-2 x^{2} \ln \relax (3)}{x \ln \relax (3) \ln \left (\frac {3}{x}\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x^{2} \ln \left (\frac {3}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x*ln(2)*ln(3)+4*x^2*ln(3))*ln(3/x)*ln(ln(3/x))+(-4*x^2*ln(2)+4*x^3+2*x)*ln(3/x)-2*ln(3))*ln((ln(3)*l
n(ln(3/x))+x)/ln(3))+(8*x^2*ln(3)*ln(2)^2-12*x^3*ln(3)*ln(2)+4*x^4*ln(3))*ln(3/x)*ln(ln(3/x))+(8*x^3*ln(2)^2+2
*(-6*x^4-2*x^2)*ln(2)+4*x^5+2*x^3)*ln(3/x)+4*x*ln(2)*ln(3)-2*x^2*ln(3))/(x*ln(3)*ln(3/x)*ln(ln(3/x))+x^2*ln(3/
x)),x)

[Out]

int((((-4*x*ln(2)*ln(3)+4*x^2*ln(3))*ln(3/x)*ln(ln(3/x))+(-4*x^2*ln(2)+4*x^3+2*x)*ln(3/x)-2*ln(3))*ln((ln(3)*l
n(ln(3/x))+x)/ln(3))+(8*x^2*ln(3)*ln(2)^2-12*x^3*ln(3)*ln(2)+4*x^4*ln(3))*ln(3/x)*ln(ln(3/x))+(8*x^3*ln(2)^2+2
*(-6*x^4-2*x^2)*ln(2)+4*x^5+2*x^3)*ln(3/x)+4*x*ln(2)*ln(3)-2*x^2*ln(3))/(x*ln(3)*ln(3/x)*ln(ln(3/x))+x^2*ln(3/
x)),x)

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maxima [B]  time = 0.50, size = 82, normalized size = 3.15 \begin {gather*} x^{4} - 4 \, x^{3} \log \relax (2) + 2 \, {\left (2 \, \log \relax (2)^{2} - \log \left (\log \relax (3)\right )\right )} x^{2} + 4 \, x \log \relax (2) \log \left (\log \relax (3)\right ) + 2 \, {\left (x^{2} - 2 \, x \log \relax (2) - \log \left (\log \relax (3)\right )\right )} \log \left (\log \relax (3) \log \left (\log \relax (3) - \log \relax (x)\right ) + x\right ) + \log \left (\log \relax (3) \log \left (\log \relax (3) - \log \relax (x)\right ) + x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(
3))*log((log(3)*log(log(3/x))+x)/log(3))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*lo
g(log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4*x*log(2)*log(3)-2*x^2*log(3))/(x*l
og(3)*log(3/x)*log(log(3/x))+x^2*log(3/x)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {3}{x}\right )\,\left (8\,x^3\,{\ln \relax (2)}^2-2\,\ln \relax (2)\,\left (6\,x^4+2\,x^2\right )+2\,x^3+4\,x^5\right )-2\,x^2\,\ln \relax (3)+\ln \left (\frac {x+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \relax (3)}{\ln \relax (3)}\right )\,\left (\ln \left (\frac {3}{x}\right )\,\left (4\,x^3-4\,\ln \relax (2)\,x^2+2\,x\right )-2\,\ln \relax (3)+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,x^2\,\ln \relax (3)-4\,x\,\ln \relax (2)\,\ln \relax (3)\right )\right )+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,\ln \relax (3)\,x^4-12\,\ln \relax (2)\,\ln \relax (3)\,x^3+8\,{\ln \relax (2)}^2\,\ln \relax (3)\,x^2\right )+4\,x\,\ln \relax (2)\,\ln \relax (3)}{x^2\,\ln \left (\frac {3}{x}\right )+x\,\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \relax (3)\,\ln \left (\frac {3}{x}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3/x)*(8*x^3*log(2)^2 - 2*log(2)*(2*x^2 + 6*x^4) + 2*x^3 + 4*x^5) - 2*x^2*log(3) + log((x + log(log(3/
x))*log(3))/log(3))*(log(3/x)*(2*x - 4*x^2*log(2) + 4*x^3) - 2*log(3) + log(log(3/x))*log(3/x)*(4*x^2*log(3) -
4*x*log(2)*log(3))) + log(log(3/x))*log(3/x)*(4*x^4*log(3) - 12*x^3*log(2)*log(3) + 8*x^2*log(2)^2*log(3)) +
4*x*log(2)*log(3))/(x^2*log(3/x) + x*log(log(3/x))*log(3)*log(3/x)),x)

[Out]

int((log(3/x)*(8*x^3*log(2)^2 - 2*log(2)*(2*x^2 + 6*x^4) + 2*x^3 + 4*x^5) - 2*x^2*log(3) + log((x + log(log(3/
x))*log(3))/log(3))*(log(3/x)*(2*x - 4*x^2*log(2) + 4*x^3) - 2*log(3) + log(log(3/x))*log(3/x)*(4*x^2*log(3) -
4*x*log(2)*log(3))) + log(log(3/x))*log(3/x)*(4*x^4*log(3) - 12*x^3*log(2)*log(3) + 8*x^2*log(2)^2*log(3)) +
4*x*log(2)*log(3))/(x^2*log(3/x) + x*log(log(3/x))*log(3)*log(3/x)), x)

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sympy [B]  time = 0.98, size = 68, normalized size = 2.62 \begin {gather*} x^{4} - 4 x^{3} \log {\relax (2 )} + 4 x^{2} \log {\relax (2 )}^{2} + \left (2 x^{2} - 4 x \log {\relax (2 )}\right ) \log {\left (\frac {x + \log {\relax (3 )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\relax (3 )}} \right )} + \log {\left (\frac {x + \log {\relax (3 )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\relax (3 )}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*ln(2)*ln(3)+4*x**2*ln(3))*ln(3/x)*ln(ln(3/x))+(-4*x**2*ln(2)+4*x**3+2*x)*ln(3/x)-2*ln(3))*ln
((ln(3)*ln(ln(3/x))+x)/ln(3))+(8*x**2*ln(3)*ln(2)**2-12*x**3*ln(3)*ln(2)+4*x**4*ln(3))*ln(3/x)*ln(ln(3/x))+(8*
x**3*ln(2)**2+2*(-6*x**4-2*x**2)*ln(2)+4*x**5+2*x**3)*ln(3/x)+4*x*ln(2)*ln(3)-2*x**2*ln(3))/(x*ln(3)*ln(3/x)*l
n(ln(3/x))+x**2*ln(3/x)),x)

[Out]

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

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