3.8.37 \(\int \frac {-x^2+x^3+x^4+9 x^6+e^{2 x} (2-3 x-9 x^3)+(-x^3-9 x^5) \log (x)}{e^{2 x} x-x^4+x^3 \log (x)} \, dx\)

Optimal. Leaf size=35 \[ -4-x+\frac {3 \left (x-x^4\right )}{x}-\log \left (\frac {e^{2 x}}{x^2}-x+\log (x)\right ) \]

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Rubi [F]  time = 2.11, 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 {-x^2+x^3+x^4+9 x^6+e^{2 x} \left (2-3 x-9 x^3\right )+\left (-x^3-9 x^5\right ) \log (x)}{e^{2 x} x-x^4+x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + x^3 + x^4 + 9*x^6 + E^(2*x)*(2 - 3*x - 9*x^3) + (-x^3 - 9*x^5)*Log[x])/(E^(2*x)*x - x^4 + x^3*Log[
x]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.41, size = 33, normalized size = 0.94 \begin {gather*} -x-3 x^3+2 \log (x)-\log \left (e^{2 x}-x^3+x^2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + x^3 + x^4 + 9*x^6 + E^(2*x)*(2 - 3*x - 9*x^3) + (-x^3 - 9*x^5)*Log[x])/(E^(2*x)*x - x^4 + x^
3*Log[x]),x]

[Out]

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

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fricas [A]  time = 0.69, size = 34, normalized size = 0.97 \begin {gather*} -3 \, x^{3} - x - \log \left (-\frac {x^{3} - x^{2} \log \relax (x) - e^{\left (2 \, x\right )}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^5-x^3)*log(x)+(-9*x^3-3*x+2)*exp(x)^2+9*x^6+x^4+x^3-x^2)/(x^3*log(x)+x*exp(x)^2-x^4),x, algor
ithm="fricas")

[Out]

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

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giac [A]  time = 0.55, size = 32, normalized size = 0.91 \begin {gather*} -3 \, x^{3} - x - \log \left (-x^{3} + x^{2} \log \relax (x) + e^{\left (2 \, x\right )}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^5-x^3)*log(x)+(-9*x^3-3*x+2)*exp(x)^2+9*x^6+x^4+x^3-x^2)/(x^3*log(x)+x*exp(x)^2-x^4),x, algor
ithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 31, normalized size = 0.89

method	result	size
risch	\(-3 x^{3}-x -\ln \left (\ln \relax (x )-\frac {x^{3}-{\mathrm e}^{2 x}}{x^{2}}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^5-x^3)*ln(x)+(-9*x^3-3*x+2)*exp(x)^2+9*x^6+x^4+x^3-x^2)/(x^3*ln(x)+x*exp(x)^2-x^4),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.63, size = 32, normalized size = 0.91 \begin {gather*} -3 \, x^{3} - x - \log \left (-x^{3} + x^{2} \log \relax (x) + e^{\left (2 \, x\right )}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^5-x^3)*log(x)+(-9*x^3-3*x+2)*exp(x)^2+9*x^6+x^4+x^3-x^2)/(x^3*log(x)+x*exp(x)^2-x^4),x, algor
ithm="maxima")

[Out]

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

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mupad [B]  time = 0.65, size = 32, normalized size = 0.91 \begin {gather*} -x-\ln \left (\frac {{\mathrm {e}}^{2\,x}+x^2\,\ln \relax (x)-x^3}{x^2}\right )-3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(3*x + 9*x^3 - 2) + log(x)*(x^3 + 9*x^5) + x^2 - x^3 - x^4 - 9*x^6)/(x*exp(2*x) + x^3*log(x) -
x^4),x)

[Out]

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

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sympy [A]  time = 0.32, size = 27, normalized size = 0.77 \begin {gather*} - 3 x^{3} - x + 2 \log {\relax (x )} - \log {\left (- x^{3} + x^{2} \log {\relax (x )} + e^{2 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**5-x**3)*ln(x)+(-9*x**3-3*x+2)*exp(x)**2+9*x**6+x**4+x**3-x**2)/(x**3*ln(x)+x*exp(x)**2-x**4)
,x)

[Out]

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

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