3.39.15 \(\int \frac {50+e^{x-x^6} (50-300 x^5)+e^x (1-x-6 e^{x-x^6} x^5)}{5000+200 e^x+2 e^{2 x}} \, dx\)

Optimal. Leaf size=28 \[ 3+\frac {e^{x-x^6}+x}{4 \left (25+\frac {e^x}{2}\right )} \]

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Rubi [F]  time = 0.97, 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 {50+e^{x-x^6} \left (50-300 x^5\right )+e^x \left (1-x-6 e^{x-x^6} x^5\right )}{5000+200 e^x+2 e^{2 x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50 + E^(x - x^6)*(50 - 300*x^5) + E^x*(1 - x - 6*E^(x - x^6)*x^5))/(5000 + 200*E^x + 2*E^(2*x)),x]

[Out]

-1/200*(1 - x)^2 - x/100 + x/(2*(50 + E^x)) + x^2/200 - ((1 - x)*Log[1 + E^x/50])/100 - (x*Log[1 + E^x/50])/10
0 + Log[50 + E^x]/100 + 25*Defer[Int][E^(x - x^6)/(50 + E^x)^2, x] - 3*Defer[Int][(E^(x - x^6)*x^5)/(50 + E^x)
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50+e^{x-x^6} \left (50-300 x^5\right )+e^x \left (1-x-6 e^{x-x^6} x^5\right )}{2 \left (50+e^x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {50+e^{x-x^6} \left (50-300 x^5\right )+e^x \left (1-x-6 e^{x-x^6} x^5\right )}{\left (50+e^x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {-50-e^x+e^x x}{\left (50+e^x\right )^2}-\frac {2 e^{x-x^6} \left (-25+150 x^5+3 e^x x^5\right )}{\left (50+e^x\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {-50-e^x+e^x x}{\left (50+e^x\right )^2} \, dx\right )-\int \frac {e^{x-x^6} \left (-25+150 x^5+3 e^x x^5\right )}{\left (50+e^x\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {-1+x}{50+e^x}-\frac {50 x}{\left (50+e^x\right )^2}\right ) \, dx\right )-\int \left (-\frac {25 e^{x-x^6}}{\left (50+e^x\right )^2}+\frac {3 e^{x-x^6} x^5}{50+e^x}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {-1+x}{50+e^x} \, dx\right )-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx+25 \int \frac {x}{\left (50+e^x\right )^2} \, dx\\ &=-\frac {1}{200} (1-x)^2+\frac {1}{100} \int \frac {e^x (-1+x)}{50+e^x} \, dx-\frac {1}{2} \int \frac {e^x x}{\left (50+e^x\right )^2} \, dx+\frac {1}{2} \int \frac {x}{50+e^x} \, dx-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx\\ &=-\frac {1}{200} (1-x)^2+\frac {x}{2 \left (50+e^x\right )}+\frac {x^2}{200}-\frac {1}{100} (1-x) \log \left (1+\frac {e^x}{50}\right )-\frac {1}{100} \int \frac {e^x x}{50+e^x} \, dx-\frac {1}{100} \int \log \left (1+\frac {e^x}{50}\right ) \, dx-\frac {1}{2} \int \frac {1}{50+e^x} \, dx-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx\\ &=-\frac {1}{200} (1-x)^2+\frac {x}{2 \left (50+e^x\right )}+\frac {x^2}{200}-\frac {1}{100} (1-x) \log \left (1+\frac {e^x}{50}\right )-\frac {1}{100} x \log \left (1+\frac {e^x}{50}\right )+\frac {1}{100} \int \log \left (1+\frac {e^x}{50}\right ) \, dx-\frac {1}{100} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{50}\right )}{x} \, dx,x,e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (50+x)} \, dx,x,e^x\right )-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx\\ &=-\frac {1}{200} (1-x)^2+\frac {x}{2 \left (50+e^x\right )}+\frac {x^2}{200}-\frac {1}{100} (1-x) \log \left (1+\frac {e^x}{50}\right )-\frac {1}{100} x \log \left (1+\frac {e^x}{50}\right )+\frac {1}{100} \text {Li}_2\left (-\frac {e^x}{50}\right )-\frac {1}{100} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\frac {1}{100} \operatorname {Subst}\left (\int \frac {1}{50+x} \, dx,x,e^x\right )+\frac {1}{100} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{50}\right )}{x} \, dx,x,e^x\right )-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx\\ &=-\frac {1}{200} (1-x)^2-\frac {x}{100}+\frac {x}{2 \left (50+e^x\right )}+\frac {x^2}{200}-\frac {1}{100} (1-x) \log \left (1+\frac {e^x}{50}\right )-\frac {1}{100} x \log \left (1+\frac {e^x}{50}\right )+\frac {1}{100} \log \left (50+e^x\right )-3 \int \frac {e^{x-x^6} x^5}{50+e^x} \, dx+25 \int \frac {e^{x-x^6}}{\left (50+e^x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 22, normalized size = 0.79 \begin {gather*} \frac {e^{x-x^6}+x}{2 \left (50+e^x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 + E^(x - x^6)*(50 - 300*x^5) + E^x*(1 - x - 6*E^(x - x^6)*x^5))/(5000 + 200*E^x + 2*E^(2*x)),x]

[Out]

(E^(x - x^6) + x)/(2*(50 + E^x))

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fricas [A]  time = 0.64, size = 18, normalized size = 0.64 \begin {gather*} \frac {x + e^{\left (-x^{6} + x\right )}}{2 \, {\left (e^{x} + 50\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5*exp(-x^6+x)-x+1)*exp(x)+(-300*x^5+50)*exp(-x^6+x)+50)/(2*exp(x)^2+200*exp(x)+5000),x, algor
ithm="fricas")

[Out]

1/2*(x + e^(-x^6 + x))/(e^x + 50)

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giac [B]  time = 0.16, size = 69, normalized size = 2.46 \begin {gather*} \frac {50 \, x e^{\left (2 \, x^{6}\right )} + x e^{\left (2 \, x^{6} + x\right )} + e^{\left (x^{6} + 2 \, x\right )} + 50 \, e^{\left (x^{6} + x\right )}}{2 \, {\left (2500 \, e^{\left (2 \, x^{6}\right )} + e^{\left (2 \, x^{6} + 2 \, x\right )} + 100 \, e^{\left (2 \, x^{6} + x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5*exp(-x^6+x)-x+1)*exp(x)+(-300*x^5+50)*exp(-x^6+x)+50)/(2*exp(x)^2+200*exp(x)+5000),x, algor
ithm="giac")

[Out]

1/2*(50*x*e^(2*x^6) + x*e^(2*x^6 + x) + e^(x^6 + 2*x) + 50*e^(x^6 + x))/(2500*e^(2*x^6) + e^(2*x^6 + 2*x) + 10
0*e^(2*x^6 + x))

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maple [A]  time = 0.08, size = 38, normalized size = 1.36

method	result	size
risch	\(\frac {x}{2 \,{\mathrm e}^{x}+100}+\frac {{\mathrm e}^{-x \left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}}{2 \,{\mathrm e}^{x}+100}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^5*exp(-x^6+x)-x+1)*exp(x)+(-300*x^5+50)*exp(-x^6+x)+50)/(2*exp(x)^2+200*exp(x)+5000),x,method=_RETU
RNVERBOSE)

[Out]

1/2*x/(exp(x)+50)+1/2/(exp(x)+50)*exp(-x*(x-1)*(x^4+x^3+x^2+x+1))

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maxima [A]  time = 0.40, size = 31, normalized size = 1.11 \begin {gather*} \frac {1}{100} \, x - \frac {x e^{x}}{100 \, {\left (e^{x} + 50\right )}} + \frac {e^{\left (-x^{6} + x\right )}}{2 \, {\left (e^{x} + 50\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5*exp(-x^6+x)-x+1)*exp(x)+(-300*x^5+50)*exp(-x^6+x)+50)/(2*exp(x)^2+200*exp(x)+5000),x, algor
ithm="maxima")

[Out]

1/100*x - 1/100*x*e^x/(e^x + 50) + 1/2*e^(-x^6 + x)/(e^x + 50)

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mupad [B]  time = 2.40, size = 28, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {e}}^{-x^6}}{2}+\frac {\frac {x}{2}-25\,{\mathrm {e}}^{-x^6}}{{\mathrm {e}}^x+50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - x^6)*(300*x^5 - 50) + exp(x)*(x + 6*x^5*exp(x - x^6) - 1) - 50)/(2*exp(2*x) + 200*exp(x) + 5000)
,x)

[Out]

exp(-x^6)/2 + (x/2 - 25*exp(-x^6))/(exp(x) + 50)

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sympy [A]  time = 0.20, size = 20, normalized size = 0.71 \begin {gather*} \frac {x}{2 e^{x} + 100} + \frac {e^{- x^{6} + x}}{2 e^{x} + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**5*exp(-x**6+x)-x+1)*exp(x)+(-300*x**5+50)*exp(-x**6+x)+50)/(2*exp(x)**2+200*exp(x)+5000),x)

[Out]

x/(2*exp(x) + 100) + exp(-x**6 + x)/(2*exp(x) + 100)

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