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

Optimal. Leaf size=33 \[ \frac {x^2}{-x^2+\log \left (-e^5+\frac {3}{x}\right )+\log \left (\frac {e^x x}{6}\right )} \]

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Rubi [F]  time = 1.59, 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 {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*Defer[Int][(x^2 - Log[-E^5 + 3/x] - Log[(E^x*x)/6])^(-2), x])/E^5 - Defer[Int][x/(x^2 - Log[-E^5 + 3/x] -
Log[(E^x*x)/6])^2, x] - Defer[Int][x^2/(x^2 - Log[-E^5 + 3/x] - Log[(E^x*x)/6])^2, x] + 2*Defer[Int][x^3/(x^2
- Log[-E^5 + 3/x] - Log[(E^x*x)/6])^2, x] - (9*Defer[Int][1/((-3 + E^5*x)*(x^2 - Log[-E^5 + 3/x] - Log[(E^x*x)
/6])^2), x])/E^5 - 2*Defer[Int][x/(x^2 - Log[-E^5 + 3/x] - Log[(E^x*x)/6]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (x \left (-3+e^5 (1+x)\right )+\left (6-2 e^5 x\right ) \log \left (-e^5+\frac {3}{x}\right )+\left (6-2 e^5 x\right ) \log \left (\frac {e^x x}{6}\right )\right )}{\left (3-e^5 x\right ) \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx\\ &=\int \left (\frac {x^2 \left (-3+e^5+\left (6+e^5\right ) x-2 e^5 x^2\right )}{\left (3-e^5 x\right ) \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}-\frac {2 x}{x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x}{x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )} \, dx\right )+\int \frac {x^2 \left (-3+e^5+\left (6+e^5\right ) x-2 e^5 x^2\right )}{\left (3-e^5 x\right ) \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {x}{x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )} \, dx\right )+\int \left (-\frac {3}{e^5 \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}-\frac {x}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}-\frac {x^2}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}+\frac {2 x^3}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}-\frac {9}{e^5 \left (-3+e^5 x\right ) \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x^3}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx-2 \int \frac {x}{x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )} \, dx-\frac {3 \int \frac {1}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx}{e^5}-\frac {9 \int \frac {1}{\left (-3+e^5 x\right ) \left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx}{e^5}-\int \frac {x}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx-\int \frac {x^2}{\left (x^2-\log \left (-e^5+\frac {3}{x}\right )-\log \left (\frac {e^x x}{6}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^2}{-x^2+\log \left (-e^5+\frac {3}{x}\right )+\log \left (\frac {e^x x}{6}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 33, normalized size = 1.00 \begin {gather*} -\frac {x^{2}}{x^{2} - \log \left (\frac {1}{6} \, x e^{x}\right ) - \log \left (-\frac {x e^{5} - 3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^
2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+((2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x
)+(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3)/x)+x^5*exp(5)-3*x^4),x, algorith
m="fricas")

[Out]

-x^2/(x^2 - log(1/6*x*e^x) - log(-(x*e^5 - 3)/x))

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giac [A]  time = 0.34, size = 26, normalized size = 0.79 \begin {gather*} -\frac {x^{2}}{x^{2} - x + \log \relax (6) - \log \left (-x e^{5} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^
2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+((2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x
)+(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3)/x)+x^5*exp(5)-3*x^4),x, algorith
m="giac")

[Out]

-x^2/(x^2 - x + log(6) - log(-x*e^5 + 3))

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maple [C]  time = 0.37, size = 239, normalized size = 7.24

method	result	size
risch	\(-\frac {2 x^{2}}{2 i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{5}-3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{5}-3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )^{3}-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 i \pi +2 x^{2}+2 \ln \relax (2)+2 \ln \relax (3)-2 \ln \left (x \,{\mathrm e}^{5}-3\right )-2 \ln \left ({\mathrm e}^{x}\right )}\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*exp(5)-6*x)*ln(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*ln((-x*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^2)/((x*e
xp(5)-3)*ln(1/6*exp(x)*x)^2+((2*x*exp(5)-6)*ln((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*ln(1/6*exp(x)*x)+(x*exp(5)
-3)*ln((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*ln((-x*exp(5)+3)/x)+x^5*exp(5)-3*x^4),x,method=_RETURNVERBOSE)

[Out]

-2*x^2/(2*I*Pi*csgn(I/x*(x*exp(5)-3))^2-I*Pi*csgn(I*(x*exp(5)-3))*csgn(I/x*(x*exp(5)-3))^2+I*Pi*csgn(I*(x*exp(
5)-3))*csgn(I/x*(x*exp(5)-3))*csgn(I/x)+I*Pi*csgn(I*x*exp(x))*csgn(I*x)*csgn(I*exp(x))-I*Pi*csgn(I*x*exp(x))^2
*csgn(I*x)-I*Pi*csgn(I/x*(x*exp(5)-3))^3-I*Pi*csgn(I*x*exp(x))^2*csgn(I*exp(x))+I*Pi*csgn(I*x*exp(x))^3-I*Pi*c
sgn(I/x*(x*exp(5)-3))^2*csgn(I/x)-2*I*Pi+2*x^2+2*ln(2)+2*ln(3)-2*ln(x*exp(5)-3)-2*ln(exp(x)))

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maxima [A]  time = 0.57, size = 28, normalized size = 0.85 \begin {gather*} -\frac {x^{2}}{x^{2} - x + \log \relax (3) + \log \relax (2) - \log \left (-x e^{5} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^
2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+((2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x
)+(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3)/x)+x^5*exp(5)-3*x^4),x, algorith
m="maxima")

[Out]

-x^2/(x^2 - x + log(3) + log(2) - log(-x*e^5 + 3))

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mupad [B]  time = 5.22, size = 30, normalized size = 0.91 \begin {gather*} \frac {x^2}{\ln \left (\frac {x\,{\mathrm {e}}^x}{6}\right )+\ln \left (-\frac {x\,{\mathrm {e}}^5-3}{x}\right )-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2/(log((x*exp(x))/6) + log(-(x*exp(5) - 3)/x) - x^2)

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sympy [A]  time = 0.51, size = 24, normalized size = 0.73 \begin {gather*} \frac {x^{2}}{- x^{2} + \log {\left (\frac {- x e^{5} + 3}{x} \right )} + \log {\left (\frac {x e^{x}}{6} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2*exp(5)-6*x)*ln(1/6*exp(x)*x)+(2*x**2*exp(5)-6*x)*ln((-x*exp(5)+3)/x)+(-x**3-x**2)*exp(5)+3*
x**2)/((x*exp(5)-3)*ln(1/6*exp(x)*x)**2+((2*x*exp(5)-6)*ln((-x*exp(5)+3)/x)-2*x**3*exp(5)+6*x**2)*ln(1/6*exp(x
)*x)+(x*exp(5)-3)*ln((-x*exp(5)+3)/x)**2+(-2*x**3*exp(5)+6*x**2)*ln((-x*exp(5)+3)/x)+x**5*exp(5)-3*x**4),x)

[Out]

x**2/(-x**2 + log((-x*exp(5) + 3)/x) + log(x*exp(x)/6))

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