3.65.90 \(\int \frac {e^{\frac {-3-4 x+\log (\frac {x+x^4-2 x^2 \log (x^2)+\log ^2(x^2)}{x})}{x}} (3 x-4 x^2+6 x^4+(4-8 x^2) \log (x^2)+2 \log ^2(x^2)+(-x-x^4+2 x^2 \log (x^2)-\log ^2(x^2)) \log (\frac {x+x^4-2 x^2 \log (x^2)+\log ^2(x^2)}{x}))}{x^3+x^6-2 x^4 \log (x^2)+x^2 \log ^2(x^2)} \, dx\)

Optimal. Leaf size=30 \[ e^{-5+\frac {-3+x+\log \left (\frac {x+\left (-x^2+\log \left (x^2\right )\right )^2}{x}\right )}{x}} \]

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Rubi [A]  time = 1.41, antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 1, number of rules used = 1, integrand size = 151, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {6706} \begin {gather*} e^{-\frac {4 x+3}{x}} \left (\frac {x^4+\log ^2\left (x^2\right )-2 x^2 \log \left (x^2\right )+x}{x}\right )^{\frac {1}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-3 - 4*x + Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x])/x)*(3*x - 4*x^2 + 6*x^4 + (4 - 8*x^2)*Log[
x^2] + 2*Log[x^2]^2 + (-x - x^4 + 2*x^2*Log[x^2] - Log[x^2]^2)*Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x])
)/(x^3 + x^6 - 2*x^4*Log[x^2] + x^2*Log[x^2]^2),x]

[Out]

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 36, normalized size = 1.20 \begin {gather*} e^{-4-\frac {3}{x}} \left (1+x^3-2 x \log \left (x^2\right )+\frac {\log ^2\left (x^2\right )}{x}\right )^{\frac {1}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-3 - 4*x + Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^2)/x])/x)*(3*x - 4*x^2 + 6*x^4 + (4 - 8*x^2
)*Log[x^2] + 2*Log[x^2]^2 + (-x - x^4 + 2*x^2*Log[x^2] - Log[x^2]^2)*Log[(x + x^4 - 2*x^2*Log[x^2] + Log[x^2]^
2)/x]))/(x^3 + x^6 - 2*x^4*Log[x^2] + x^2*Log[x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.69, size = 38, normalized size = 1.27 \begin {gather*} e^{\left (-\frac {4 \, x - \log \left (\frac {x^{4} - 2 \, x^{2} \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + x}{x}\right ) + 3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.09, size = 35, normalized size = 1.17 \begin {gather*} e^{\left (\frac {\log \left (x^{3} - 2 \, x \log \left (x^{2}\right ) + \frac {\log \left (x^{2}\right )^{2}}{x} + 1\right )}{x} - \frac {3}{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-\ln \left (x^{2}\right )^{2}+2 x^{2} \ln \left (x^{2}\right )-x^{4}-x \right ) \ln \left (\frac {\ln \left (x^{2}\right )^{2}-2 x^{2} \ln \left (x^{2}\right )+x^{4}+x}{x}\right )+2 \ln \left (x^{2}\right )^{2}+\left (-8 x^{2}+4\right ) \ln \left (x^{2}\right )+6 x^{4}-4 x^{2}+3 x \right ) {\mathrm e}^{\frac {\ln \left (\frac {\ln \left (x^{2}\right )^{2}-2 x^{2} \ln \left (x^{2}\right )+x^{4}+x}{x}\right )-4 x -3}{x}}}{x^{2} \ln \left (x^{2}\right )^{2}-2 x^{4} \ln \left (x^{2}\right )+x^{6}+x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 38, normalized size = 1.27 \begin {gather*} e^{\left (\frac {\log \left (x^{4} - 4 \, x^{2} \log \relax (x) + 4 \, \log \relax (x)^{2} + x\right )}{x} - \frac {\log \relax (x)}{x} - \frac {3}{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.43, size = 35, normalized size = 1.17 \begin {gather*} {\mathrm {e}}^{-4}\,{\mathrm {e}}^{-\frac {3}{x}}\,{\left (x^3-2\,x\,\ln \left (x^2\right )+\frac {{\ln \left (x^2\right )}^2}{x}+1\right )}^{1/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.06, size = 32, normalized size = 1.07 \begin {gather*} e^{\frac {- 4 x + \log {\left (\frac {x^{4} - 2 x^{2} \log {\left (x^{2} \right )} + x + \log {\left (x^{2} \right )}^{2}}{x} \right )} - 3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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