3.25.92 \(\int \frac {4-2 x+4 x^3+e^x (2-2 x^2)+e^{3 x} (4+e^x (2-8 x)-12 x+8 x^2+6 x^3)+(-4-2 x^2+e^x (-2+2 x)) \log (x)}{-2 x^2-e^x x^2+x^3+x^4+e^{3 x} (-2 x-e^x x+x^2+x^3)+(2 x+e^x x-x^2-x^3) \log (x)} \, dx\)

Optimal. Leaf size=31 \[ \log \left (\frac {\left (-2-e^x+x+x^2\right )^2 \left (e^{3 x}+x-\log (x)\right )^2}{x^2}\right ) \]

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Rubi [F]  time = 2.55, 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 {4-2 x+4 x^3+e^x \left (2-2 x^2\right )+e^{3 x} \left (4+e^x (2-8 x)-12 x+8 x^2+6 x^3\right )+\left (-4-2 x^2+e^x (-2+2 x)\right ) \log (x)}{-2 x^2-e^x x^2+x^3+x^4+e^{3 x} \left (-2 x-e^x x+x^2+x^3\right )+\left (2 x+e^x x-x^2-x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - 2*x + 4*x^3 + E^x*(2 - 2*x^2) + E^(3*x)*(4 + E^x*(2 - 8*x) - 12*x + 8*x^2 + 6*x^3) + (-4 - 2*x^2 + E^
x*(-2 + 2*x))*Log[x])/(-2*x^2 - E^x*x^2 + x^3 + x^4 + E^(3*x)*(-2*x - E^x*x + x^2 + x^3) + (2*x + E^x*x - x^2
- x^3)*Log[x]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 33, normalized size = 1.06 \begin {gather*} 2 \left (-\log (x)+\log \left (2+e^x-x-x^2\right )+\log \left (e^{3 x}+x-\log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 2*x + 4*x^3 + E^x*(2 - 2*x^2) + E^(3*x)*(4 + E^x*(2 - 8*x) - 12*x + 8*x^2 + 6*x^3) + (-4 - 2*x^
2 + E^x*(-2 + 2*x))*Log[x])/(-2*x^2 - E^x*x^2 + x^3 + x^4 + E^(3*x)*(-2*x - E^x*x + x^2 + x^3) + (2*x + E^x*x
- x^2 - x^3)*Log[x]),x]

[Out]

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

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fricas [A]  time = 0.66, size = 35, normalized size = 1.13 \begin {gather*} 2 \, \log \left (-x^{2} - x + e^{x} + 2\right ) - 2 \, \log \relax (x) + 2 \, \log \left (-x - e^{\left (3 \, x\right )} + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-2)*exp(x)-2*x^2-4)*log(x)+((-8*x+2)*exp(x)+6*x^3+8*x^2-12*x+4)*exp(3*x)+(-2*x^2+2)*exp(x)+4*x
^3-2*x+4)/((exp(x)*x-x^3-x^2+2*x)*log(x)+(-exp(x)*x+x^3+x^2-2*x)*exp(3*x)-exp(x)*x^2+x^4+x^3-2*x^2),x, algorit
hm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-2)*exp(x)-2*x^2-4)*log(x)+((-8*x+2)*exp(x)+6*x^3+8*x^2-12*x+4)*exp(3*x)+(-2*x^2+2)*exp(x)+4*x
^3-2*x+4)/((exp(x)*x-x^3-x^2+2*x)*log(x)+(-exp(x)*x+x^3+x^2-2*x)*exp(3*x)-exp(x)*x^2+x^4+x^3-2*x^2),x, algorit
hm="giac")

[Out]

Timed out

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maple [A]  time = 0.04, size = 36, normalized size = 1.16

method	result	size
risch	\(-2 \ln \relax (x )+2 \ln \left (2-x -x^{2}+{\mathrm e}^{x}\right )+2 \ln \left (-{\mathrm e}^{3 x}-x +\ln \relax (x )\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x-2)*exp(x)-2*x^2-4)*ln(x)+((-8*x+2)*exp(x)+6*x^3+8*x^2-12*x+4)*exp(3*x)+(-2*x^2+2)*exp(x)+4*x^3-2*x+
4)/((exp(x)*x-x^3-x^2+2*x)*ln(x)+(-exp(x)*x+x^3+x^2-2*x)*exp(3*x)-exp(x)*x^2+x^4+x^3-2*x^2),x,method=_RETURNVE
RBOSE)

[Out]

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

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maxima [A]  time = 0.52, size = 33, normalized size = 1.06 \begin {gather*} 2 \, \log \left (-x^{2} - x + e^{x} + 2\right ) + 2 \, \log \left (x + e^{\left (3 \, x\right )} - \log \relax (x)\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-2)*exp(x)-2*x^2-4)*log(x)+((-8*x+2)*exp(x)+6*x^3+8*x^2-12*x+4)*exp(3*x)+(-2*x^2+2)*exp(x)+4*x
^3-2*x+4)/((exp(x)*x-x^3-x^2+2*x)*log(x)+(-exp(x)*x+x^3+x^2-2*x)*exp(3*x)-exp(x)*x^2+x^4+x^3-2*x^2),x, algorit
hm="maxima")

[Out]

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

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mupad [B]  time = 1.58, size = 31, normalized size = 1.00 \begin {gather*} 2\,\ln \left (x-{\mathrm {e}}^x+x^2-2\right )+2\,\ln \left (x+{\mathrm {e}}^{3\,x}-\ln \relax (x)\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + log(x)*(2*x^2 - exp(x)*(2*x - 2) + 4) - exp(3*x)*(8*x^2 - exp(x)*(8*x - 2) - 12*x + 6*x^3 + 4) + ex
p(x)*(2*x^2 - 2) - 4*x^3 - 4)/(x^2*exp(x) - log(x)*(2*x + x*exp(x) - x^2 - x^3) + 2*x^2 - x^3 - x^4 + exp(3*x)
*(2*x + x*exp(x) - x^2 - x^3)),x)

[Out]

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

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sympy [B]  time = 2.88, size = 60, normalized size = 1.94 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (- x^{3} + x^{2} \log {\relax (x )} - x^{2} + x \log {\relax (x )} + 2 x + \left (x - \log {\relax (x )}\right ) e^{x} + \left (- x^{2} - x + 2\right ) e^{3 x} + e^{4 x} - 2 \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-2)*exp(x)-2*x**2-4)*ln(x)+((-8*x+2)*exp(x)+6*x**3+8*x**2-12*x+4)*exp(3*x)+(-2*x**2+2)*exp(x)+
4*x**3-2*x+4)/((exp(x)*x-x**3-x**2+2*x)*ln(x)+(-exp(x)*x+x**3+x**2-2*x)*exp(3*x)-exp(x)*x**2+x**4+x**3-2*x**2)
,x)

[Out]

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

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