3.71.33 \(\int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x (-9 x^2+6 x^3-x^4)}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x (9 x^3-6 x^4+x^5)} \, dx\)

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

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Rubi [F]  time = 3.05, 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 {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + 2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 + E^x*(-9*x^2 + 6*x^3 - x^4))/(6*x - 2*x^2 + 45*x^3 - 3
9*x^4 + 11*x^5 - x^6 + E^x*(9*x^3 - 6*x^4 + x^5)),x]

[Out]

-Log[x] + 2*Defer[Int][(2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x^4)^(-1), x] + 2*Defer[Int][1/((-3 + x)*(2
+ 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x^4)), x] + 4*Defer[Int][1/(x*(2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*
x^3 + x^4)), x] + 18*Defer[Int][x^2/(2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x^4), x] - 9*Defer[Int][x^3/(2
+ 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x^4), x] + Defer[Int][x^4/(2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3
+ x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{(3-x) x \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {-12+2 x^2-54 x^3+45 x^4-12 x^5+x^6}{(-3+x) x \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )}\right ) \, dx\\ &=-\log (x)+\int \frac {-12+2 x^2-54 x^3+45 x^4-12 x^5+x^6}{(-3+x) x \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )} \, dx\\ &=-\log (x)+\int \left (\frac {2}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4}+\frac {2}{(-3+x) \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )}+\frac {4}{x \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )}+\frac {18 x^2}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4}-\frac {9 x^3}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4}+\frac {x^4}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4}\right ) \, dx\\ &=-\log (x)+2 \int \frac {1}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4} \, dx+2 \int \frac {1}{(-3+x) \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )} \, dx+4 \int \frac {1}{x \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right )} \, dx-9 \int \frac {x^3}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4} \, dx+18 \int \frac {x^2}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4} \, dx+\int \frac {x^4}{2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 44, normalized size = 1.33 \begin {gather*} x+\log (3-x)+\log (x)-\log \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 + E^x*(-9*x^2 + 6*x^3 - x^4))/(6*x - 2*x^2 + 45*x
^3 - 39*x^4 + 11*x^5 - x^6 + E^x*(9*x^3 - 6*x^4 + x^5)),x]

[Out]

x + Log[3 - x] + Log[x] - Log[2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x^4]

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fricas [A]  time = 0.58, size = 50, normalized size = 1.52 \begin {gather*} x - \log \relax (x) - \log \left (-\frac {x^{4} - 8 \, x^{3} + 15 \, x^{2} - {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + 2}{x^{3} - 3 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x
^5-39*x^4+45*x^3-2*x^2+6*x),x, algorithm="fricas")

[Out]

x - log(x) - log(-(x^4 - 8*x^3 + 15*x^2 - (x^3 - 3*x^2)*e^x + 2)/(x^3 - 3*x^2))

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giac [A]  time = 0.16, size = 41, normalized size = 1.24 \begin {gather*} x - \log \left (-x^{4} + x^{3} e^{x} + 8 \, x^{3} - 3 \, x^{2} e^{x} - 15 \, x^{2} - 2\right ) + \log \left (x - 3\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x
^5-39*x^4+45*x^3-2*x^2+6*x),x, algorithm="giac")

[Out]

x - log(-x^4 + x^3*e^x + 8*x^3 - 3*x^2*e^x - 15*x^2 - 2) + log(x - 3) + log(x)

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maple [A]  time = 0.06, size = 38, normalized size = 1.15

method	result	size
risch	\(-\ln \relax (x )+x -\ln \left ({\mathrm e}^{x}-\frac {x^{4}-8 x^{3}+15 x^{2}+2}{x^{2} \left (x -3\right )}\right )\)	\(38\)
norman	\(x -\ln \left (x^{4}-{\mathrm e}^{x} x^{3}-8 x^{3}+3 \,{\mathrm e}^{x} x^{2}+15 x^{2}+2\right )+\ln \relax (x )+\ln \left (x -3\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*
x^4+45*x^3-2*x^2+6*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+x-ln(exp(x)-(x^4-8*x^3+15*x^2+2)/x^2/(x-3))

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maxima [A]  time = 0.43, size = 50, normalized size = 1.52 \begin {gather*} x - \log \relax (x) - \log \left (-\frac {x^{4} - 8 \, x^{3} + 15 \, x^{2} - {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + 2}{x^{3} - 3 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x
^5-39*x^4+45*x^3-2*x^2+6*x),x, algorithm="maxima")

[Out]

x - log(x) - log(-(x^4 - 8*x^3 + 15*x^2 - (x^3 - 3*x^2)*e^x + 2)/(x^3 - 3*x^2))

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mupad [B]  time = 4.18, size = 40, normalized size = 1.21 \begin {gather*} x-\ln \left (3\,x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+15\,x^2-8\,x^3+x^4+2\right )+\ln \left (x-3\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 - exp(x)*(9*x^2 - 6*x^3 + x^4) + 6)/(6*x - 2*x^2 + 45*x^3 -
39*x^4 + 11*x^5 - x^6 + exp(x)*(9*x^3 - 6*x^4 + x^5)),x)

[Out]

x - log(3*x^2*exp(x) - x^3*exp(x) + 15*x^2 - 8*x^3 + x^4 + 2) + log(x - 3) + log(x)

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sympy [A]  time = 0.43, size = 32, normalized size = 0.97 \begin {gather*} x - \log {\relax (x )} - \log {\left (e^{x} + \frac {- x^{4} + 8 x^{3} - 15 x^{2} - 2}{x^{3} - 3 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4+6*x**3-9*x**2)*exp(x)-x**6+13*x**5-56*x**4+93*x**3-47*x**2+2*x+6)/((x**5-6*x**4+9*x**3)*exp(
x)-x**6+11*x**5-39*x**4+45*x**3-2*x**2+6*x),x)

[Out]

x - log(x) - log(exp(x) + (-x**4 + 8*x**3 - 15*x**2 - 2)/(x**3 - 3*x**2))

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