3.21.41 \(\int \frac {-60 x+15 x^2+e^{-e^4+x} (30 x^2+15 x^3-15 x^4)+e^x (-3+x+e^{-e^4+x} (2+x-x^2))}{e^x (-2+x)-30 x^2+15 x^3} \, dx\)

Optimal. Leaf size=37 \[ x-e^{-e^4+x} x-\log \left (\frac {-2+x}{\frac {1}{3}+5 e^{-x} x^2}\right ) \]

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Rubi [F]  time = 1.06, 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 {-60 x+15 x^2+e^{-e^4+x} \left (30 x^2+15 x^3-15 x^4\right )+e^x \left (-3+x+e^{-e^4+x} \left (2+x-x^2\right )\right )}{e^x (-2+x)-30 x^2+15 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-60*x + 15*x^2 + E^(-E^4 + x)*(30*x^2 + 15*x^3 - 15*x^4) + E^x*(-3 + x + E^(-E^4 + x)*(2 + x - x^2)))/(E^
x*(-2 + x) - 30*x^2 + 15*x^3),x]

[Out]

E^(-E^4 + x) + x - E^(-E^4 + x)*(1 + x) - Log[2 - x] + 30*Defer[Int][x/(E^x + 15*x^2), x] - 15*Defer[Int][x^2/
(E^x + 15*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {60 x-15 x^2-e^{-e^4+x} \left (30 x^2+15 x^3-15 x^4\right )-e^x \left (-3+x+e^{-e^4+x} \left (2+x-x^2\right )\right )}{(2-x) \left (e^x+15 x^2\right )} \, dx\\ &=\int \left (\frac {-3+x}{-2+x}-e^{-e^4+x} (1+x)-\frac {15 (-2+x) x}{e^x+15 x^2}\right ) \, dx\\ &=-\left (15 \int \frac {(-2+x) x}{e^x+15 x^2} \, dx\right )+\int \frac {-3+x}{-2+x} \, dx-\int e^{-e^4+x} (1+x) \, dx\\ &=-e^{-e^4+x} (1+x)-15 \int \left (-\frac {2 x}{e^x+15 x^2}+\frac {x^2}{e^x+15 x^2}\right ) \, dx+\int e^{-e^4+x} \, dx+\int \left (1+\frac {1}{2-x}\right ) \, dx\\ &=e^{-e^4+x}+x-e^{-e^4+x} (1+x)-\log (2-x)-15 \int \frac {x^2}{e^x+15 x^2} \, dx+30 \int \frac {x}{e^x+15 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 46, normalized size = 1.24 \begin {gather*} e^{-e^4} \left (-e^x x-e^{e^4} \log \left (1-\frac {x}{2}\right )+e^{e^4} \log \left (e^x+15 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-60*x + 15*x^2 + E^(-E^4 + x)*(30*x^2 + 15*x^3 - 15*x^4) + E^x*(-3 + x + E^(-E^4 + x)*(2 + x - x^2)
))/(E^x*(-2 + x) - 30*x^2 + 15*x^3),x]

[Out]

(-(E^x*x) - E^E^4*Log[1 - x/2] + E^E^4*Log[E^x + 15*x^2])/E^E^4

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fricas [A]  time = 0.57, size = 34, normalized size = 0.92 \begin {gather*} -{\left (x e^{x} - e^{\left (e^{4}\right )} \log \left (15 \, x^{2} + e^{x}\right ) + e^{\left (e^{4}\right )} \log \left (x - 2\right )\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+2)*exp(x-exp(4))+x-3)*exp(x)+(-15*x^4+15*x^3+30*x^2)*exp(x-exp(4))+15*x^2-60*x)/(exp(x)*(x
-2)+15*x^3-30*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.32, size = 34, normalized size = 0.92 \begin {gather*} -{\left (x e^{x} - e^{\left (e^{4}\right )} \log \left (15 \, x^{2} + e^{x}\right ) + e^{\left (e^{4}\right )} \log \left (x - 2\right )\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+2)*exp(x-exp(4))+x-3)*exp(x)+(-15*x^4+15*x^3+30*x^2)*exp(x-exp(4))+15*x^2-60*x)/(exp(x)*(x
-2)+15*x^3-30*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.11, size = 27, normalized size = 0.73

method	result	size
norman	\(-{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{4}} x -\ln \left (x -2\right )+\ln \left (15 x^{2}+{\mathrm e}^{x}\right )\)	\(27\)
risch	\(-{\mathrm e}^{x -{\mathrm e}^{4}} x -\ln \left (x -2\right )+\ln \left (15 x^{2}+{\mathrm e}^{x}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+x+2)*exp(x-exp(4))+x-3)*exp(x)+(-15*x^4+15*x^3+30*x^2)*exp(x-exp(4))+15*x^2-60*x)/(exp(x)*(x-2)+15
*x^3-30*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.61, size = 26, normalized size = 0.70 \begin {gather*} -x e^{\left (x - e^{4}\right )} + \log \left (15 \, x^{2} + e^{x}\right ) - \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+2)*exp(x-exp(4))+x-3)*exp(x)+(-15*x^4+15*x^3+30*x^2)*exp(x-exp(4))+15*x^2-60*x)/(exp(x)*(x
-2)+15*x^3-30*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.18, size = 26, normalized size = 0.70 \begin {gather*} \ln \left ({\mathrm {e}}^x+15\,x^2\right )-\ln \left (x-2\right )-x\,{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - exp(4))*(30*x^2 + 15*x^3 - 15*x^4) - 60*x + exp(x)*(x + exp(x - exp(4))*(x - x^2 + 2) - 3) + 15*x
^2)/(exp(x)*(x - 2) - 30*x^2 + 15*x^3),x)

[Out]

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

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sympy [A]  time = 0.21, size = 24, normalized size = 0.65 \begin {gather*} - \frac {x e^{x}}{e^{e^{4}}} - \log {\left (x - 2 \right )} + \log {\left (15 x^{2} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+x+2)*exp(x-exp(4))+x-3)*exp(x)+(-15*x**4+15*x**3+30*x**2)*exp(x-exp(4))+15*x**2-60*x)/(exp(
x)*(x-2)+15*x**3-30*x**2),x)

[Out]

-x*exp(x)*exp(-exp(4)) - log(x - 2) + log(15*x**2 + exp(x))

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