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

Optimal. Leaf size=34 \[ -x+\log (4)-e^2 \left (-2-x+\frac {x}{-1+\log \left (\frac {3+\frac {2}{x^2}}{x}\right )}\right ) \]

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Rubi [F]  time = 0.87, 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 {-2-3 x^2+e^2 \left (-2+3 x^2\right )+\left (4+6 x^2+e^2 \left (-6-9 x^2\right )\right ) \log \left (\frac {2+3 x^2}{x^3}\right )+\left (-2-3 x^2+e^2 \left (2+3 x^2\right )\right ) \log ^2\left (\frac {2+3 x^2}{x^3}\right )}{2+3 x^2+\left (-4-6 x^2\right ) \log \left (\frac {2+3 x^2}{x^3}\right )+\left (2+3 x^2\right ) \log ^2\left (\frac {2+3 x^2}{x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((1 - E^2)*x) - E^2*Defer[Int][(-1 + Log[(2 + 3*x^2)/x^3])^(-2), x] - I*Sqrt[2]*E^2*Defer[Int][1/((I*Sqrt[2]
- Sqrt[3]*x)*(-1 + Log[(2 + 3*x^2)/x^3])^2), x] - I*Sqrt[2]*E^2*Defer[Int][1/((I*Sqrt[2] + Sqrt[3]*x)*(-1 + Lo
g[(2 + 3*x^2)/x^3])^2), x] - E^2*Defer[Int][(-1 + Log[(2 + 3*x^2)/x^3])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2-3 x^2+e^2 \left (-2+3 x^2\right )+\left (4+6 x^2+e^2 \left (-6-9 x^2\right )\right ) \log \left (\frac {2+3 x^2}{x^3}\right )+\left (-2-3 x^2+e^2 \left (2+3 x^2\right )\right ) \log ^2\left (\frac {2+3 x^2}{x^3}\right )}{\left (2+3 x^2\right ) \left (1-\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx\\ &=\int \left (-1+e^2-\frac {3 e^2 \left (2+x^2\right )}{\left (2+3 x^2\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2}-\frac {e^2}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )}\right ) \, dx\\ &=-\left (\left (1-e^2\right ) x\right )-e^2 \int \frac {1}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \, dx-\left (3 e^2\right ) \int \frac {2+x^2}{\left (2+3 x^2\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx\\ &=-\left (\left (1-e^2\right ) x\right )-e^2 \int \frac {1}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \, dx-\left (3 e^2\right ) \int \left (\frac {1}{3 \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2}+\frac {4}{3 \left (2+3 x^2\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2}\right ) \, dx\\ &=-\left (\left (1-e^2\right ) x\right )-e^2 \int \frac {1}{\left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx-e^2 \int \frac {1}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \, dx-\left (4 e^2\right ) \int \frac {1}{\left (2+3 x^2\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx\\ &=-\left (\left (1-e^2\right ) x\right )-e^2 \int \frac {1}{\left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx-e^2 \int \frac {1}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \, dx-\left (4 e^2\right ) \int \left (\frac {i}{2 \sqrt {2} \left (i \sqrt {2}-\sqrt {3} x\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2}+\frac {i}{2 \sqrt {2} \left (i \sqrt {2}+\sqrt {3} x\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2}\right ) \, dx\\ &=-\left (\left (1-e^2\right ) x\right )-e^2 \int \frac {1}{\left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx-e^2 \int \frac {1}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \, dx-\left (i \sqrt {2} e^2\right ) \int \frac {1}{\left (i \sqrt {2}-\sqrt {3} x\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx-\left (i \sqrt {2} e^2\right ) \int \frac {1}{\left (i \sqrt {2}+\sqrt {3} x\right ) \left (-1+\log \left (\frac {2+3 x^2}{x^3}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 30, normalized size = 0.88 \begin {gather*} \left (-1+e^2\right ) x-\frac {e^2 x}{-1+\log \left (\frac {2+3 x^2}{x^3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + E^2)*x - (E^2*x)/(-1 + Log[(2 + 3*x^2)/x^3])

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fricas [A]  time = 0.56, size = 49, normalized size = 1.44 \begin {gather*} -\frac {2 \, x e^{2} - {\left (x e^{2} - x\right )} \log \left (\frac {3 \, x^{2} + 2}{x^{3}}\right ) - x}{\log \left (\frac {3 \, x^{2} + 2}{x^{3}}\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+2)*exp(2)-3*x^2-2)*log((3*x^2+2)/x^3)^2+((-9*x^2-6)*exp(2)+6*x^2+4)*log((3*x^2+2)/x^3)+(3*x
^2-2)*exp(2)-3*x^2-2)/((3*x^2+2)*log((3*x^2+2)/x^3)^2+(-6*x^2-4)*log((3*x^2+2)/x^3)+3*x^2+2),x, algorithm="fri
cas")

[Out]

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

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giac [A]  time = 0.45, size = 55, normalized size = 1.62 \begin {gather*} \frac {x e^{2} \log \left (\frac {3 \, x^{2} + 2}{x^{3}}\right ) - 2 \, x e^{2} - x \log \left (\frac {3 \, x^{2} + 2}{x^{3}}\right ) + x}{\log \left (\frac {3 \, x^{2} + 2}{x^{3}}\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+2)*exp(2)-3*x^2-2)*log((3*x^2+2)/x^3)^2+((-9*x^2-6)*exp(2)+6*x^2+4)*log((3*x^2+2)/x^3)+(3*x
^2-2)*exp(2)-3*x^2-2)/((3*x^2+2)*log((3*x^2+2)/x^3)^2+(-6*x^2-4)*log((3*x^2+2)/x^3)+3*x^2+2),x, algorithm="gia
c")

[Out]

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

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maple [A]  time = 0.08, size = 30, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{2} x -x -\frac {{\mathrm e}^{2} x}{\ln \left (\frac {3 x^{2}+2}{x^{3}}\right )-1}\)	\(30\)
norman	\(\frac {\left (-2 \,{\mathrm e}^{2}+1\right ) x +\left ({\mathrm e}^{2}-1\right ) x \ln \left (\frac {3 x^{2}+2}{x^{3}}\right )}{\ln \left (\frac {3 x^{2}+2}{x^{3}}\right )-1}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^2+2)*exp(2)-3*x^2-2)*ln((3*x^2+2)/x^3)^2+((-9*x^2-6)*exp(2)+6*x^2+4)*ln((3*x^2+2)/x^3)+(3*x^2-2)*ex
p(2)-3*x^2-2)/((3*x^2+2)*ln((3*x^2+2)/x^3)^2+(-6*x^2-4)*ln((3*x^2+2)/x^3)+3*x^2+2),x,method=_RETURNVERBOSE)

[Out]

exp(2)*x-x-exp(2)*x/(ln((3*x^2+2)/x^3)-1)

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maxima [A]  time = 0.44, size = 50, normalized size = 1.47 \begin {gather*} \frac {x {\left (e^{2} - 1\right )} \log \left (3 \, x^{2} + 2\right ) - 3 \, x {\left (e^{2} - 1\right )} \log \relax (x) - x {\left (2 \, e^{2} - 1\right )}}{\log \left (3 \, x^{2} + 2\right ) - 3 \, \log \relax (x) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+2)*exp(2)-3*x^2-2)*log((3*x^2+2)/x^3)^2+((-9*x^2-6)*exp(2)+6*x^2+4)*log((3*x^2+2)/x^3)+(3*x
^2-2)*exp(2)-3*x^2-2)/((3*x^2+2)*log((3*x^2+2)/x^3)^2+(-6*x^2-4)*log((3*x^2+2)/x^3)+3*x^2+2),x, algorithm="max
ima")

[Out]

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

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mupad [B]  time = 1.32, size = 53, normalized size = 1.56 \begin {gather*} -\frac {x\,\left (\ln \left (\frac {3\,x^2+2}{x^3}\right )+2\,{\mathrm {e}}^2-\ln \left (\frac {3\,x^2+2}{x^3}\right )\,{\mathrm {e}}^2-1\right )}{\ln \left (\frac {3\,x^2+2}{x^3}\right )-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(log((3*x^2 + 2)/x^3) + 2*exp(2) - log((3*x^2 + 2)/x^3)*exp(2) - 1))/(log((3*x^2 + 2)/x^3) - 1)

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sympy [A]  time = 0.19, size = 24, normalized size = 0.71 \begin {gather*} x \left (-1 + e^{2}\right ) - \frac {x e^{2}}{\log {\left (\frac {3 x^{2} + 2}{x^{3}} \right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-1 + exp(2)) - x*exp(2)/(log((3*x**2 + 2)/x**3) - 1)

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