3.91.24 \(\int \frac {(-2 x \log ^2(3)+e^{2+x-x^4} (-2+4 x-16 x^4) \log ^2(3)) \log (\frac {e^{4+2 x-2 x^4}-2 e^{2+x-x^4} x+x^2}{x})}{e^{2+x-x^4} x-x^2} \, dx\)

Optimal. Leaf size=35 \[ \log ^2(3) \log ^2\left (\frac {\left (e^{x \left (-x^3+\frac {2+x}{x}\right )}-x\right )^2}{x}\right ) \]

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Rubi [A]  time = 2.34, antiderivative size = 41, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 2, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6742, 6686} \begin {gather*} \log ^2(3) \log ^2\left (\frac {-2 e^{-x^4+x+2} x+e^{-2 x^4+2 x+4}+x^2}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2*x*Log[3]^2 + E^(2 + x - x^4)*(-2 + 4*x - 16*x^4)*Log[3]^2)*Log[(E^(4 + 2*x - 2*x^4) - 2*E^(2 + x - x^
4)*x + x^2)/x])/(E^(2 + x - x^4)*x - x^2),x]

[Out]

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

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 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.51, size = 35, normalized size = 1.00 \begin {gather*} \log ^2(3) \log ^2\left (\frac {e^{-2 x^4} \left (e^{2+x}-e^{x^4} x\right )^2}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2*x*Log[3]^2 + E^(2 + x - x^4)*(-2 + 4*x - 16*x^4)*Log[3]^2)*Log[(E^(4 + 2*x - 2*x^4) - 2*E^(2 +
x - x^4)*x + x^2)/x])/(E^(2 + x - x^4)*x - x^2),x]

[Out]

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

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fricas [A]  time = 0.48, size = 39, normalized size = 1.11 \begin {gather*} \log \relax (3)^{2} \log \left (\frac {x^{2} - 2 \, x e^{\left (-x^{4} + x + 2\right )} + e^{\left (-2 \, x^{4} + 2 \, x + 4\right )}}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+4*x-2)*log(3)^2*exp(-x^4+x+2)-2*x*log(3)^2)*log((exp(-x^4+x+2)^2-2*x*exp(-x^4+x+2)+x^2)/x)
/(x*exp(-x^4+x+2)-x^2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (8 \, x^{4} - 2 \, x + 1\right )} e^{\left (-x^{4} + x + 2\right )} \log \relax (3)^{2} + x \log \relax (3)^{2}\right )} \log \left (\frac {x^{2} - 2 \, x e^{\left (-x^{4} + x + 2\right )} + e^{\left (-2 \, x^{4} + 2 \, x + 4\right )}}{x}\right )}{x^{2} - x e^{\left (-x^{4} + x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+4*x-2)*log(3)^2*exp(-x^4+x+2)-2*x*log(3)^2)*log((exp(-x^4+x+2)^2-2*x*exp(-x^4+x+2)+x^2)/x)
/(x*exp(-x^4+x+2)-x^2),x, algorithm="giac")

[Out]

integrate(2*((8*x^4 - 2*x + 1)*e^(-x^4 + x + 2)*log(3)^2 + x*log(3)^2)*log((x^2 - 2*x*e^(-x^4 + x + 2) + e^(-2
*x^4 + 2*x + 4))/x)/(x^2 - x*e^(-x^4 + x + 2)), x)

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maple [A]  time = 0.51, size = 40, normalized size = 1.14

method	result	size
norman	\(\ln \relax (3)^{2} \ln \left (\frac {{\mathrm e}^{-2 x^{4}+2 x +4}-2 x \,{\mathrm e}^{-x^{4}+x +2}+x^{2}}{x}\right )^{2}\)	\(40\)
risch	\(-32 \ln \relax (3)^{2}+4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{3}+4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{3}+\left (-4 \ln \relax (3)^{2} \ln \relax (x )-16 \ln \relax (3)^{2}\right ) \ln \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )+\ln \relax (3)^{2} \ln \relax (x )^{2}+16 \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right )+4 \ln \relax (3)^{2} \ln \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}-2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )+i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )+2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}-2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right )^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )+i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right )^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )-i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}-4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}+4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right )^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )-8 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{2}-4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}-2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{3}-i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}-2 i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{2}+4 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{2}+2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{2}+4 i \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )-2 i \pi \ln \relax (3)^{2} \ln \left ({\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}-x \right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{3}+i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}\right )^{3}+i \ln \relax (x ) \pi \ln \relax (3)^{2} \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\left (x +1\right ) \left (x^{3}-x^{2}+x -2\right )}+x \right )^{2}}{x}\right )^{3}\)	\(1442\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^4+4*x-2)*ln(3)^2*exp(-x^4+x+2)-2*x*ln(3)^2)*ln((exp(-x^4+x+2)^2-2*x*exp(-x^4+x+2)+x^2)/x)/(x*exp(-
x^4+x+2)-x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left ({\left (8 \, x^{4} - 2 \, x + 1\right )} e^{\left (-x^{4} + x + 2\right )} \log \relax (3)^{2} + x \log \relax (3)^{2}\right )} \log \left (\frac {x^{2} - 2 \, x e^{\left (-x^{4} + x + 2\right )} + e^{\left (-2 \, x^{4} + 2 \, x + 4\right )}}{x}\right )}{x^{2} - x e^{\left (-x^{4} + x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4+4*x-2)*log(3)^2*exp(-x^4+x+2)-2*x*log(3)^2)*log((exp(-x^4+x+2)^2-2*x*exp(-x^4+x+2)+x^2)/x)
/(x*exp(-x^4+x+2)-x^2),x, algorithm="maxima")

[Out]

2*integrate(((8*x^4 - 2*x + 1)*e^(-x^4 + x + 2)*log(3)^2 + x*log(3)^2)*log((x^2 - 2*x*e^(-x^4 + x + 2) + e^(-2
*x^4 + 2*x + 4))/x)/(x^2 - x*e^(-x^4 + x + 2)), x)

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mupad [B]  time = 8.79, size = 38, normalized size = 1.09 \begin {gather*} {\ln \relax (3)}^2\,{\ln \left (x-2\,{\mathrm {e}}^2\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^x+\frac {{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-2\,x^4}}{x}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((exp(2*x - 2*x^4 + 4) - 2*x*exp(x - x^4 + 2) + x^2)/x)*(2*x*log(3)^2 + exp(x - x^4 + 2)*log(3)^2*(16
*x^4 - 4*x + 2)))/(x*exp(x - x^4 + 2) - x^2),x)

[Out]

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

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sympy [A]  time = 0.58, size = 36, normalized size = 1.03 \begin {gather*} \log {\relax (3 )}^{2} \log {\left (\frac {x^{2} - 2 x e^{- x^{4} + x + 2} + e^{- 2 x^{4} + 2 x + 4}}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**4+4*x-2)*ln(3)**2*exp(-x**4+x+2)-2*x*ln(3)**2)*ln((exp(-x**4+x+2)**2-2*x*exp(-x**4+x+2)+x**
2)/x)/(x*exp(-x**4+x+2)-x**2),x)

[Out]

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

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