3.45.68 \(\int \frac {-9-18 x+e^{x^2} (1+2 x-8 x^2-2 x^3-2 x^4)+(-36-9 x-9 x^2+e^{x^2} (4+x+x^2)) \log (\frac {2}{-9+e^{x^2}})}{-36-9 x-9 x^2+e^{x^2} (4+x+x^2)} \, dx\)

Optimal. Leaf size=22 \[ x \log \left (\frac {2}{-9+e^{x^2}}\right )+\log \left (4+x+x^2\right ) \]

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Rubi [F]  time = 1.26, 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 {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9 - 18*x + E^x^2*(1 + 2*x - 8*x^2 - 2*x^3 - 2*x^4) + (-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2))*Log[2/(-9
+ E^x^2)])/(-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2)),x]

[Out]

(-2*x^3)/3 + x*Log[-2/(9 - E^x^2)] + Log[4 + x + x^2] - 18*Defer[Int][x^2/(-9 + E^x^2), x] + 2*Defer[Int][(E^x
^2*x^2)/(-9 + E^x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9+18 x-e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )-\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{\left (9-e^{x^2}\right ) \left (4+x+x^2\right )} \, dx\\ &=\int \left (-\frac {18 x^2}{-9+e^{x^2}}+\frac {1+2 x-8 x^2-2 x^3-2 x^4+4 \log \left (\frac {2}{-9+e^{x^2}}\right )+x \log \left (\frac {2}{-9+e^{x^2}}\right )+x^2 \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2}\right ) \, dx\\ &=-\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4+4 \log \left (\frac {2}{-9+e^{x^2}}\right )+x \log \left (\frac {2}{-9+e^{x^2}}\right )+x^2 \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2} \, dx\\ &=-\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4+\left (4+x+x^2\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2} \, dx\\ &=-\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \left (\frac {1+2 x-8 x^2-2 x^3-2 x^4}{4+x+x^2}+\log \left (\frac {2}{-9+e^{x^2}}\right )\right ) \, dx\\ &=-\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4}{4+x+x^2} \, dx+\int \log \left (\frac {2}{-9+e^{x^2}}\right ) \, dx\\ &=x \log \left (-\frac {2}{9-e^{x^2}}\right )-18 \int \frac {x^2}{-9+e^{x^2}} \, dx-\int -\frac {2 e^{x^2} x^2}{-9+e^{x^2}} \, dx+\int \left (-2 x^2+\frac {1+2 x}{4+x+x^2}\right ) \, dx\\ &=-\frac {2 x^3}{3}+x \log \left (-\frac {2}{9-e^{x^2}}\right )+2 \int \frac {e^{x^2} x^2}{-9+e^{x^2}} \, dx-18 \int \frac {x^2}{-9+e^{x^2}} \, dx+\int \frac {1+2 x}{4+x+x^2} \, dx\\ &=-\frac {2 x^3}{3}+x \log \left (-\frac {2}{9-e^{x^2}}\right )+\log \left (4+x+x^2\right )+2 \int \frac {e^{x^2} x^2}{-9+e^{x^2}} \, dx-18 \int \frac {x^2}{-9+e^{x^2}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 22, normalized size = 1.00 \begin {gather*} x \log \left (\frac {2}{-9+e^{x^2}}\right )+\log \left (4+x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 18*x + E^x^2*(1 + 2*x - 8*x^2 - 2*x^3 - 2*x^4) + (-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2))*Log
[2/(-9 + E^x^2)])/(-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2)),x]

[Out]

x*Log[2/(-9 + E^x^2)] + Log[4 + x + x^2]

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fricas [A]  time = 0.48, size = 21, normalized size = 0.95 \begin {gather*} x \log \left (\frac {2}{e^{\left (x^{2}\right )} - 9}\right ) + \log \left (x^{2} + x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="fricas")

[Out]

x*log(2/(e^(x^2) - 9)) + log(x^2 + x + 4)

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giac [A]  time = 0.21, size = 21, normalized size = 0.95 \begin {gather*} x \log \left (\frac {2}{e^{\left (x^{2}\right )} - 9}\right ) + \log \left (x^{2} + x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="giac")

[Out]

x*log(2/(e^(x^2) - 9)) + log(x^2 + x + 4)

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maple [A]  time = 0.18, size = 22, normalized size = 1.00

method	result	size
norman	\(\ln \left (\frac {2}{{\mathrm e}^{x^{2}}-9}\right ) x +\ln \left (x^{2}+x +4\right )\)	\(22\)
risch	\(-\ln \left ({\mathrm e}^{x^{2}}-9\right ) x +x \ln \relax (2)+\ln \left (x^{2}+x +4\right )\)	\(23\)
default	\(x \left (\ln \left (\frac {2}{{\mathrm e}^{x^{2}}-9}\right )+\ln \left ({\mathrm e}^{x^{2}}-9\right )\right )-\ln \left ({\mathrm e}^{x^{2}}-9\right ) x +\ln \left (x^{2}+x +4\right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*ln(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/((x^2+x+
4)*exp(x^2)-9*x^2-9*x-36),x,method=_RETURNVERBOSE)

[Out]

ln(2/(exp(x^2)-9))*x+ln(x^2+x+4)

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maxima [A]  time = 0.48, size = 22, normalized size = 1.00 \begin {gather*} x \log \relax (2) - x \log \left (e^{\left (x^{2}\right )} - 9\right ) + \log \left (x^{2} + x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="maxima")

[Out]

x*log(2) - x*log(e^(x^2) - 9) + log(x^2 + x + 4)

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mupad [B]  time = 3.50, size = 21, normalized size = 0.95 \begin {gather*} \ln \left (x^2+x+4\right )+x\,\ln \left (\frac {2}{{\mathrm {e}}^{x^2}-9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + exp(x^2)*(8*x^2 - 2*x + 2*x^3 + 2*x^4 - 1) + log(2/(exp(x^2) - 9))*(9*x - exp(x^2)*(x + x^2 + 4) +
9*x^2 + 36) + 9)/(9*x - exp(x^2)*(x + x^2 + 4) + 9*x^2 + 36),x)

[Out]

log(x + x^2 + 4) + x*log(2/(exp(x^2) - 9))

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sympy [A]  time = 0.36, size = 19, normalized size = 0.86 \begin {gather*} x \log {\left (\frac {2}{e^{x^{2}} - 9} \right )} + \log {\left (x^{2} + x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+x+4)*exp(x**2)-9*x**2-9*x-36)*ln(2/(exp(x**2)-9))+(-2*x**4-2*x**3-8*x**2+2*x+1)*exp(x**2)-18
*x-9)/((x**2+x+4)*exp(x**2)-9*x**2-9*x-36),x)

[Out]

x*log(2/(exp(x**2) - 9)) + log(x**2 + x + 4)

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