3.38.65 \(\int \frac {-2 x^2+(-2-8 x^2-10 x^4-4 x^6) \log (\frac {e^{-x^2}}{x})+(4 x^2+4 x^4) \log ^2(\frac {e^{-x^2}}{x})}{x} \, dx\) [3765]

Optimal. Leaf size=33 \[ e^{e^4}-x^2+\left (1+x^2\right )^2 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \]

[Out]

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

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Rubi [F]
time = 0.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^2 + 4*x^4)*Log[1/(E^x^2*x)]^2)/x,x]

[Out]

-3*x^2 - (21*x^4)/8 - (17*x^6)/18 - x^8/6 - 4*x^2*Log[1/(E^x^2*x)] - (5*x^4*Log[1/(E^x^2*x)])/2 - (2*x^6*Log[1
/(E^x^2*x)])/3 - 2*Defer[Int][Log[1/(E^x^2*x)]/x, x] + 4*Defer[Int][x*Log[1/(E^x^2*x)]^2, x] + 4*Defer[Int][x^
3*Log[1/(E^x^2*x)]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 x-\frac {2 \left (1+x^2\right )^2 \left (1+2 x^2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x}+4 x \left (1+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )\right ) \, dx\\ &=-x^2-2 \int \frac {\left (1+x^2\right )^2 \left (1+2 x^2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+4 \int x \left (1+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-2 \int \left (\frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x}+4 x \log \left (\frac {e^{-x^2}}{x}\right )+5 x^3 \log \left (\frac {e^{-x^2}}{x}\right )+2 x^5 \log \left (\frac {e^{-x^2}}{x}\right )\right ) \, dx+4 \int \left (x \log ^2\left (\frac {e^{-x^2}}{x}\right )+x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right )\right ) \, dx\\ &=-x^2-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx-4 \int x^5 \log \left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx-8 \int x \log \left (\frac {e^{-x^2}}{x}\right ) \, dx-10 \int x^3 \log \left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )+\frac {2}{3} \int x^5 \left (-1-2 x^2\right ) \, dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+\frac {5}{2} \int x^3 \left (-1-2 x^2\right ) \, dx+4 \int x \left (-1-2 x^2\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )+\frac {2}{3} \int \left (-x^5-2 x^7\right ) \, dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+\frac {5}{2} \int \left (-x^3-2 x^5\right ) \, dx+4 \int \left (-x-2 x^3\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-3 x^2-\frac {21 x^4}{8}-\frac {17 x^6}{18}-\frac {x^8}{6}-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).
time = 0.06, size = 72, normalized size = 2.18 \begin {gather*} \log ^2\left (\frac {1}{x}\right )+x^2 \left (2+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+2 \log \left (\frac {1}{x}\right ) \log (x)-2 \log \left (\frac {e^{-x^2}}{x}\right ) \left (x^2+\log (x)\right )-x^2 \left (1+x^2+2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^2 + 4*x^4)*Log[1/(E^x^2*x)]^2)/x,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(30)=60\).
time = 1.95, size = 490, normalized size = 14.85

method	result	size
default	\(\frac {4 x^{6} \ln \left (x \right )}{3}+x^{4} \ln \left (x \right )^{2}+\frac {3 x^{4} \ln \left (x \right )}{2}+2 x^{2} \ln \left (x \right )^{2}-4 x^{2} \ln \left (x \right )-\frac {x^{6}}{2}-\ln \left (x \right )^{2}-3 x^{4}-x^{2}-\frac {65}{144}+2 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{4} \ln \left (x \right )+4 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{2} \ln \left (x \right )-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \left (x \right )+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )+\frac {5 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4}}{2}+\frac {2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{6}}{3}+x^{4} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )^{2}+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )^{2}-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 x^{2} \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{2} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}+\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )^{2}-\frac {\left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{4}}{2}-2 x^{2} \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}\)	\(490\)
risch	\(\text {Expression too large to display}\)	\(7067\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+4*x^2)*ln(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*ln(1/exp(x^2)/x)-2*x^2)/x,x,method=_RETURNVERBOS
E)

[Out]

4/3*x^6*ln(x)+x^4*ln(x)^2+3/2*x^4*ln(x)+2*x^2*ln(x)^2-4*x^2*ln(x)-1/2*x^6-ln(x)^2-3*x^4-x^2-65/144+(ln(1/exp(x
^2)/x)+ln(exp(x^2))+ln(x))^2+2*(ln(exp(x^2))-x^2)*x^4*ln(x)+4*(ln(exp(x^2))-x^2)*x^2*ln(x)-2*(ln(1/exp(x^2)/x)
+ln(exp(x^2))+ln(x))*x^4*ln(x)-4*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))*x^2*ln(x)-4*(ln(1/exp(x^2)/x)+ln(exp(x^
2))+ln(x))*ln(exp(x^2))*x^2-2*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))*ln(exp(x^2))*x^4-2*ln(1/exp(x^2)/x)*ln(x)-
1/2*(ln(exp(x^2))-x^2)*x^4-2*x^2*(ln(exp(x^2))-x^2)+ln(exp(x^2))^2*x^4+2*ln(exp(x^2))^2*x^2-2/3*ln(exp(x^2))*x
^6-2*ln(exp(x^2))*x^4+2*x^2*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))+5/2*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))*x^
4+2/3*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))*x^6+x^4*(ln(1/exp(x^2)/x)+ln(exp(x^2))+ln(x))^2+2*x^2*(ln(1/exp(x^
2)/x)+ln(exp(x^2))+ln(x))^2-5/2*ln(1/exp(x^2)/x)*x^4-4*x^2*ln(1/exp(x^2)/x)-2/3*ln(1/exp(x^2)/x)*x^6

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (30) = 60\).
time = 0.30, size = 171, normalized size = 5.18 \begin {gather*} -\frac {2}{3} \, x^{6} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - \frac {5}{2} \, x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{4} + 2 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - 4 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{2} - {\left (2 \, x^{2} + \log \left (x^{2}\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} + \frac {1}{6} \, {\left (4 \, x^{6} + 3 \, x^{4}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + 2 \, {\left (x^{4} + x^{2}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - 2 \, \log \left (x\right ) \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1/exp(x^2)/x)-2*x^2)/x,x, algorithm="
maxima")

[Out]

-2/3*x^6*log(e^(-x^2)/x) + x^4*log(e^(-x^2)/x)^2 - 5/2*x^4*log(e^(-x^2)/x) - x^4 + 2*x^2*log(e^(-x^2)/x)^2 - 4
*x^2*log(e^(-x^2)/x) - x^2 - (2*x^2 + log(x^2))*log(x) + log(x)^2 + 1/6*(4*x^6 + 3*x^4)*log(e^(-x^2)/x) + 2*(x
^4 + x^2)*log(e^(-x^2)/x) - 2*log(x)*log(e^(-x^2)/x)

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Fricas [A]
time = 0.36, size = 30, normalized size = 0.91 \begin {gather*} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1/exp(x^2)/x)-2*x^2)/x,x, algorithm="
fricas")

[Out]

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

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Sympy [A]
time = 0.12, size = 24, normalized size = 0.73 \begin {gather*} - x^{2} + \left (x^{4} + 2 x^{2} + 1\right ) \log {\left (\frac {e^{- x^{2}}}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+4*x**2)*ln(1/exp(x**2)/x)**2+(-4*x**6-10*x**4-8*x**2-2)*ln(1/exp(x**2)/x)-2*x**2)/x,x)

[Out]

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

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Giac [A]
time = 0.40, size = 48, normalized size = 1.45 \begin {gather*} x^{8} + 2 \, x^{6} + x^{4} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1/exp(x^2)/x)-2*x^2)/x,x, algorithm="
giac")

[Out]

x^8 + 2*x^6 + x^4 + (x^4 + 2*x^2 + 1)*log(x)^2 - x^2 + 2*(x^6 + 2*x^4 + x^2)*log(x)

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Mupad [B]
time = 2.22, size = 30, normalized size = 0.91 \begin {gather*} {\ln \left (\frac {{\mathrm {e}}^{-x^2}}{x}\right )}^2\,\left (x^4+2\,x^2+1\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(-x^2)/x)*(8*x^2 + 10*x^4 + 4*x^6 + 2) - log(exp(-x^2)/x)^2*(4*x^2 + 4*x^4) + 2*x^2)/x,x)

[Out]

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

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