3.14.94 \(\int \frac {e^{-e^{9-x}} (-40+10 x+e^{9-x} (122-80 x+10 x^2)+e^{9-x} (61-40 x+5 x^2) \log (61-40 x+5 x^2))}{61-40 x+5 x^2} \, dx\)

Optimal. Leaf size=30 \[ e^{-e^{9-x}} \left (2+\log \left (6-5 \left (5-(4-x)^2\right )\right )\right ) \]

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Rubi [B]  time = 0.36, antiderivative size = 76, normalized size of antiderivative = 2.53, number of steps used = 1, number of rules used = 1, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {2288} \begin {gather*} \frac {e^{x-e^{9-x}-9} \left (2 e^{9-x} \left (5 x^2-40 x+61\right )+e^{9-x} \left (5 x^2-40 x+61\right ) \log \left (5 x^2-40 x+61\right )\right )}{5 x^2-40 x+61} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-40 + 10*x + E^(9 - x)*(122 - 80*x + 10*x^2) + E^(9 - x)*(61 - 40*x + 5*x^2)*Log[61 - 40*x + 5*x^2])/(E^E
^(9 - x)*(61 - 40*x + 5*x^2)),x]

[Out]

(E^(-9 - E^(9 - x) + x)*(2*E^(9 - x)*(61 - 40*x + 5*x^2) + E^(9 - x)*(61 - 40*x + 5*x^2)*Log[61 - 40*x + 5*x^2
]))/(61 - 40*x + 5*x^2)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-9-e^{9-x}+x} \left (2 e^{9-x} \left (61-40 x+5 x^2\right )+e^{9-x} \left (61-40 x+5 x^2\right ) \log \left (61-40 x+5 x^2\right )\right )}{61-40 x+5 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 25, normalized size = 0.83 \begin {gather*} e^{-e^{9-x}} \left (2+\log \left (61-40 x+5 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40 + 10*x + E^(9 - x)*(122 - 80*x + 10*x^2) + E^(9 - x)*(61 - 40*x + 5*x^2)*Log[61 - 40*x + 5*x^2]
)/(E^E^(9 - x)*(61 - 40*x + 5*x^2)),x]

[Out]

(2 + Log[61 - 40*x + 5*x^2])/E^E^(9 - x)

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fricas [A]  time = 0.53, size = 23, normalized size = 0.77 \begin {gather*} {\left (\log \left (5 \, x^{2} - 40 \, x + 61\right ) + 2\right )} e^{\left (-e^{\left (-x + 9\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-40*x+61)*exp(9-x)*log(5*x^2-40*x+61)+(10*x^2-80*x+122)*exp(9-x)+10*x-40)/(5*x^2-40*x+61)/exp
(exp(9-x)),x, algorithm="fricas")

[Out]

(log(5*x^2 - 40*x + 61) + 2)*e^(-e^(-x + 9))

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giac [B]  time = 0.46, size = 48, normalized size = 1.60 \begin {gather*} {\left (e^{\left (-x - e^{\left (-x + 9\right )} + 9\right )} \log \left (5 \, x^{2} - 40 \, x + 61\right ) + 2 \, e^{\left (-x - e^{\left (-x + 9\right )} + 9\right )}\right )} e^{\left (x - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-40*x+61)*exp(9-x)*log(5*x^2-40*x+61)+(10*x^2-80*x+122)*exp(9-x)+10*x-40)/(5*x^2-40*x+61)/exp
(exp(9-x)),x, algorithm="giac")

[Out]

(e^(-x - e^(-x + 9) + 9)*log(5*x^2 - 40*x + 61) + 2*e^(-x - e^(-x + 9) + 9))*e^(x - 9)

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maple [A]  time = 0.38, size = 24, normalized size = 0.80

method	result	size
risch	\(\left (2+\ln \left (5 x^{2}-40 x +61\right )\right ) {\mathrm e}^{-{\mathrm e}^{9-x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^2-40*x+61)*exp(9-x)*ln(5*x^2-40*x+61)+(10*x^2-80*x+122)*exp(9-x)+10*x-40)/(5*x^2-40*x+61)/exp(exp(9-
x)),x,method=_RETURNVERBOSE)

[Out]

(2+ln(5*x^2-40*x+61))*exp(-exp(9-x))

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maxima [A]  time = 0.65, size = 23, normalized size = 0.77 \begin {gather*} {\left (\log \left (5 \, x^{2} - 40 \, x + 61\right ) + 2\right )} e^{\left (-e^{\left (-x + 9\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-40*x+61)*exp(9-x)*log(5*x^2-40*x+61)+(10*x^2-80*x+122)*exp(9-x)+10*x-40)/(5*x^2-40*x+61)/exp
(exp(9-x)),x, algorithm="maxima")

[Out]

(log(5*x^2 - 40*x + 61) + 2)*e^(-e^(-x + 9))

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mupad [B]  time = 0.17, size = 23, normalized size = 0.77 \begin {gather*} {\mathrm {e}}^{-{\mathrm {e}}^{-x}\,{\mathrm {e}}^9}\,\left (\ln \left (5\,x^2-40\,x+61\right )+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(9 - x))*(10*x + exp(9 - x)*(10*x^2 - 80*x + 122) + exp(9 - x)*log(5*x^2 - 40*x + 61)*(5*x^2 - 40
*x + 61) - 40))/(5*x^2 - 40*x + 61),x)

[Out]

exp(-exp(-x)*exp(9))*(log(5*x^2 - 40*x + 61) + 2)

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sympy [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(((5*x**2-40*x+61)*exp(9-x)*ln(5*x**2-40*x+61)+(10*x**2-80*x+122)*exp(9-x)+10*x-40)/(5*x**2-40*x+61)/
exp(exp(9-x)),x)

[Out]

Timed out

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