3.103.75 \(\int \frac {e^{-e^x} ((20+x^2) \log (2)+e^x (-20 x+x^2+x^3) \log (2))}{2000-200 x-195 x^2+10 x^3+5 x^4} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^{-e^x} x \log (2)}{5 (4-x) (5+x)} \]

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Rubi [A]  time = 0.11, antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 1, number of rules used = 1, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {e^{-e^x} \left (-x^3-x^2+20 x\right ) \log (2)}{5 \left (x^4+2 x^3-39 x^2-40 x+400\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((20 + x^2)*Log[2] + E^x*(-20*x + x^2 + x^3)*Log[2])/(E^E^x*(2000 - 200*x - 195*x^2 + 10*x^3 + 5*x^4)),x]

[Out]

((20*x - x^2 - x^3)*Log[2])/(5*E^E^x*(400 - 40*x - 39*x^2 + 2*x^3 + x^4))

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^{-e^x} \left (20 x-x^2-x^3\right ) \log (2)}{5 \left (400-40 x-39 x^2+2 x^3+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.85 \begin {gather*} -\frac {e^{-e^x} x \log (2)}{5 \left (-20+x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((20 + x^2)*Log[2] + E^x*(-20*x + x^2 + x^3)*Log[2])/(E^E^x*(2000 - 200*x - 195*x^2 + 10*x^3 + 5*x^4
)),x]

[Out]

-1/5*(x*Log[2])/(E^E^x*(-20 + x + x^2))

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fricas [A]  time = 0.68, size = 18, normalized size = 0.69 \begin {gather*} -\frac {x e^{\left (-e^{x}\right )} \log \relax (2)}{5 \, {\left (x^{2} + x - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-20*x)*log(2)*exp(x)+(x^2+20)*log(2))/(5*x^4+10*x^3-195*x^2-200*x+2000)/exp(exp(x)),x, algo
rithm="fricas")

[Out]

-1/5*x*e^(-e^x)*log(2)/(x^2 + x - 20)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{3} + x^{2} - 20 \, x\right )} e^{x} \log \relax (2) + {\left (x^{2} + 20\right )} \log \relax (2)\right )} e^{\left (-e^{x}\right )}}{5 \, {\left (x^{4} + 2 \, x^{3} - 39 \, x^{2} - 40 \, x + 400\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-20*x)*log(2)*exp(x)+(x^2+20)*log(2))/(5*x^4+10*x^3-195*x^2-200*x+2000)/exp(exp(x)),x, algo
rithm="giac")

[Out]

integrate(1/5*((x^3 + x^2 - 20*x)*e^x*log(2) + (x^2 + 20)*log(2))*e^(-e^x)/(x^4 + 2*x^3 - 39*x^2 - 40*x + 400)
, x)

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maple [A]  time = 0.13, size = 19, normalized size = 0.73

method	result	size
norman	\(-\frac {x \ln \relax (2) {\mathrm e}^{-{\mathrm e}^{x}}}{5 \left (x^{2}+x -20\right )}\)	\(19\)
risch	\(-\frac {x \ln \relax (2) {\mathrm e}^{-{\mathrm e}^{x}}}{5 \left (x^{2}+x -20\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+x^2-20*x)*ln(2)*exp(x)+(x^2+20)*ln(2))/(5*x^4+10*x^3-195*x^2-200*x+2000)/exp(exp(x)),x,method=_RETUR
NVERBOSE)

[Out]

-1/5*x*ln(2)/(x^2+x-20)/exp(exp(x))

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maxima [A]  time = 0.49, size = 18, normalized size = 0.69 \begin {gather*} -\frac {x e^{\left (-e^{x}\right )} \log \relax (2)}{5 \, {\left (x^{2} + x - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-20*x)*log(2)*exp(x)+(x^2+20)*log(2))/(5*x^4+10*x^3-195*x^2-200*x+2000)/exp(exp(x)),x, algo
rithm="maxima")

[Out]

-1/5*x*e^(-e^x)*log(2)/(x^2 + x - 20)

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mupad [B]  time = 6.85, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,\ln \relax (2)}{5\,\left (x^2+x-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(x))*(log(2)*(x^2 + 20) + exp(x)*log(2)*(x^2 - 20*x + x^3)))/(10*x^3 - 195*x^2 - 200*x + 5*x^4 +
2000),x)

[Out]

-(x*exp(-exp(x))*log(2))/(5*(x + x^2 - 20))

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sympy [A]  time = 0.31, size = 20, normalized size = 0.77 \begin {gather*} - \frac {x e^{- e^{x}} \log {\relax (2 )}}{5 x^{2} + 5 x - 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+x**2-20*x)*ln(2)*exp(x)+(x**2+20)*ln(2))/(5*x**4+10*x**3-195*x**2-200*x+2000)/exp(exp(x)),x)

[Out]

-x*exp(-exp(x))*log(2)/(5*x**2 + 5*x - 100)

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