3.44.44 \(\int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} (16-16 x^2+x^3-4 x^4+2 x^5)}{8 x^4+2 x^6-x^7+e^{x^2} (-8 x-2 x^3+x^4)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\left (2+\frac {8}{x^2}-x\right ) \left (-e^{x^2}+x^3\right )\right ) \]

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Rubi [F]  time = 1.33, 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 {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x^3 + 6*x^5 - 4*x^6 + E^x^2*(16 - 16*x^2 + x^3 - 4*x^4 + 2*x^5))/(8*x^4 + 2*x^6 - x^7 + E^x^2*(-8*x - 2
*x^3 + x^4)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 x^3-6 x^5+4 x^6-e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{x \left (e^{x^2}-x^3\right ) \left (8+2 x^2-x^3\right )} \, dx\\ &=\int \left (-\frac {x^2 \left (-3+2 x^2\right )}{-e^{x^2}+x^3}+\frac {16-16 x^2+x^3-4 x^4+2 x^5}{x \left (-8-2 x^2+x^3\right )}\right ) \, dx\\ &=-\int \frac {x^2 \left (-3+2 x^2\right )}{-e^{x^2}+x^3} \, dx+\int \frac {16-16 x^2+x^3-4 x^4+2 x^5}{x \left (-8-2 x^2+x^3\right )} \, dx\\ &=-\int \left (-\frac {3 x^2}{-e^{x^2}+x^3}+\frac {2 x^4}{-e^{x^2}+x^3}\right ) \, dx+\int \left (-\frac {2}{x}+2 x+\frac {x (-4+3 x)}{-8-2 x^2+x^3}\right ) \, dx\\ &=x^2-2 \log (x)-2 \int \frac {x^4}{-e^{x^2}+x^3} \, dx+3 \int \frac {x^2}{-e^{x^2}+x^3} \, dx+\int \frac {x (-4+3 x)}{-8-2 x^2+x^3} \, dx\\ &=x^2-2 \log (x)+\log \left (8+2 x^2-x^3\right )-2 \int \frac {x^4}{-e^{x^2}+x^3} \, dx+3 \int \frac {x^2}{-e^{x^2}+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 30, normalized size = 1.30 \begin {gather*} -2 \log (x)+\log \left (e^{x^2}-x^3\right )+\log \left (8+2 x^2-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^3 + 6*x^5 - 4*x^6 + E^x^2*(16 - 16*x^2 + x^3 - 4*x^4 + 2*x^5))/(8*x^4 + 2*x^6 - x^7 + E^x^2*(-8
*x - 2*x^3 + x^4)),x]

[Out]

-2*Log[x] + Log[E^x^2 - x^3] + Log[8 + 2*x^2 - x^3]

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fricas [A]  time = 0.64, size = 27, normalized size = 1.17 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),
x, algorithm="fricas")

[Out]

log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)

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giac [A]  time = 0.13, size = 27, normalized size = 1.17 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),
x, algorithm="giac")

[Out]

log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)

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maple [A]  time = 0.07, size = 28, normalized size = 1.22

method	result	size
norman	\(-2 \ln \relax (x )+\ln \left (x^{3}-{\mathrm e}^{x^{2}}\right )+\ln \left (x^{3}-2 x^{2}-8\right )\)	\(28\)
risch	\(-2 \ln \relax (x )+\ln \left (x^{3}-2 x^{2}-8\right )+\ln \left (-x^{3}+{\mathrm e}^{x^{2}}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),x,meth
od=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(x^3-exp(x^2))+ln(x^3-2*x^2-8)

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maxima [A]  time = 0.46, size = 27, normalized size = 1.17 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),
x, algorithm="maxima")

[Out]

log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)

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mupad [B]  time = 3.26, size = 27, normalized size = 1.17 \begin {gather*} \ln \left (x^3-{\mathrm {e}}^{x^2}\right )+\ln \left (x^3-2\,x^2-8\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(x^3 - 16*x^2 - 4*x^4 + 2*x^5 + 16) + 8*x^3 + 6*x^5 - 4*x^6)/(exp(x^2)*(8*x + 2*x^3 - x^4) - 8*
x^4 - 2*x^6 + x^7),x)

[Out]

log(x^3 - exp(x^2)) + log(x^3 - 2*x^2 - 8) - 2*log(x)

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sympy [A]  time = 0.20, size = 26, normalized size = 1.13 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (- x^{3} + e^{x^{2}} \right )} + \log {\left (x^{3} - 2 x^{2} - 8 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5-4*x**4+x**3-16*x**2+16)*exp(x**2)-4*x**6+6*x**5+8*x**3)/((x**4-2*x**3-8*x)*exp(x**2)-x**7+2
*x**6+8*x**4),x)

[Out]

-2*log(x) + log(-x**3 + exp(x**2)) + log(x**3 - 2*x**2 - 8)

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