3.10.68 \(\int \frac {5+e^{e^5 x-x^2} (1+e^5 (-1-x)+2 x+2 x^2)}{25+e^{2 e^5 x-2 x^2}+10 e^{e^5 x-x^2}} \, dx\)

Optimal. Leaf size=21 \[ 2+\frac {1+x}{5+e^{\left (e^5-x\right ) x}} \]

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Rubi [F]  time = 3.14, 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 {5+e^{e^5 x-x^2} \left (1+e^5 (-1-x)+2 x+2 x^2\right )}{25+e^{2 e^5 x-2 x^2}+10 e^{e^5 x-x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 + E^(E^5*x - x^2)*(1 + E^5*(-1 - x) + 2*x + 2*x^2))/(25 + E^(2*E^5*x - 2*x^2) + 10*E^(E^5*x - x^2)),x]

[Out]

(E^(-(E^5*x) + x^2)*(E^5 - (2 - E^5)*x - 2*x^2))/(E^5 - 2*x) + 5*Defer[Int][E^(5 + 2*x^2)/(E^(E^5*x) + 5*E^x^2
)^2, x] - 5*(1 - E^5)*Defer[Int][E^(2*x^2)/(E^(2*E^5*x) + 5*E^(x*(E^5 + x))), x] - 5*(2 - E^5)*Defer[Int][(E^(
2*x^2)*x)/(E^(E^5*x) + 5*E^x^2)^2, x] - 5*(2 - E^5)*Defer[Int][(E^(2*x^2)*x)/(E^(2*E^5*x) + 5*E^(x*(E^5 + x)))
, x] - 10*Defer[Int][(E^(2*x^2)*x^2)/(E^(E^5*x) + 5*E^x^2)^2, x] - 10*Defer[Int][(E^(2*x^2)*x^2)/(E^(2*E^5*x)
+ 5*E^(x*(E^5 + x))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x^2} \left (5+e^{e^5 x-x^2} \left (1+e^5 (-1-x)+2 x+2 x^2\right )\right )}{\left (e^{e^5 x}+5 e^{x^2}\right )^2} \, dx\\ &=\int \left (\frac {5 e^{2 x^2} \left (e^5-2 x\right ) (1+x)}{\left (e^{e^5 x}+5 e^{x^2}\right )^2}+\frac {5 e^{2 x^2} \left (-1+e^5-\left (2-e^5\right ) x-2 x^2\right )}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}}+e^{-e^5 x+x^2} \left (1-e^5+\left (2-e^5\right ) x+2 x^2\right )\right ) \, dx\\ &=5 \int \frac {e^{2 x^2} \left (e^5-2 x\right ) (1+x)}{\left (e^{e^5 x}+5 e^{x^2}\right )^2} \, dx+5 \int \frac {e^{2 x^2} \left (-1+e^5-\left (2-e^5\right ) x-2 x^2\right )}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}} \, dx+\int e^{-e^5 x+x^2} \left (1-e^5+\left (2-e^5\right ) x+2 x^2\right ) \, dx\\ &=\frac {e^{-e^5 x+x^2} \left (e^5-\left (2-e^5\right ) x-2 x^2\right )}{e^5-2 x}+5 \int \left (\frac {e^{5+2 x^2}}{\left (e^{e^5 x}+5 e^{x^2}\right )^2}+\frac {e^{2 x^2} \left (-2+e^5\right ) x}{\left (e^{e^5 x}+5 e^{x^2}\right )^2}-\frac {2 e^{2 x^2} x^2}{\left (e^{e^5 x}+5 e^{x^2}\right )^2}\right ) \, dx+5 \int \left (-\frac {e^{2 x^2} \left (1-e^5\right )}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}}+\frac {e^{2 x^2} \left (-2+e^5\right ) x}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}}-\frac {2 e^{2 x^2} x^2}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}}\right ) \, dx\\ &=\frac {e^{-e^5 x+x^2} \left (e^5-\left (2-e^5\right ) x-2 x^2\right )}{e^5-2 x}+5 \int \frac {e^{5+2 x^2}}{\left (e^{e^5 x}+5 e^{x^2}\right )^2} \, dx-10 \int \frac {e^{2 x^2} x^2}{\left (e^{e^5 x}+5 e^{x^2}\right )^2} \, dx-10 \int \frac {e^{2 x^2} x^2}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}} \, dx-\left (5 \left (1-e^5\right )\right ) \int \frac {e^{2 x^2}}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}} \, dx-\left (5 \left (2-e^5\right )\right ) \int \frac {e^{2 x^2} x}{\left (e^{e^5 x}+5 e^{x^2}\right )^2} \, dx-\left (5 \left (2-e^5\right )\right ) \int \frac {e^{2 x^2} x}{e^{2 e^5 x}+5 e^{x \left (e^5+x\right )}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 37, normalized size = 1.76 \begin {gather*} \frac {x}{5}-\frac {e^{e^5 x} (1+x)}{5 \left (e^{e^5 x}+5 e^{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + E^(E^5*x - x^2)*(1 + E^5*(-1 - x) + 2*x + 2*x^2))/(25 + E^(2*E^5*x - 2*x^2) + 10*E^(E^5*x - x^2
)),x]

[Out]

x/5 - (E^(E^5*x)*(1 + x))/(5*(E^(E^5*x) + 5*E^x^2))

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fricas [A]  time = 0.68, size = 19, normalized size = 0.90 \begin {gather*} \frac {x + 1}{e^{\left (-x^{2} + x e^{5}\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(5)+2*x^2+2*x+1)*exp(x*exp(5)-x^2)+5)/(exp(x*exp(5)-x^2)^2+10*exp(x*exp(5)-x^2)+25),x, a
lgorithm="fricas")

[Out]

(x + 1)/(e^(-x^2 + x*e^5) + 5)

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giac [A]  time = 0.45, size = 19, normalized size = 0.90 \begin {gather*} \frac {x + 1}{e^{\left (-x^{2} + x e^{5}\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(5)+2*x^2+2*x+1)*exp(x*exp(5)-x^2)+5)/(exp(x*exp(5)-x^2)^2+10*exp(x*exp(5)-x^2)+25),x, a
lgorithm="giac")

[Out]

(x + 1)/(e^(-x^2 + x*e^5) + 5)

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maple [A]  time = 0.10, size = 18, normalized size = 0.86

method	result	size
risch	\(\frac {x +1}{5+{\mathrm e}^{x \left ({\mathrm e}^{5}-x \right )}}\)	\(18\)
norman	\(\frac {x +1}{{\mathrm e}^{x \,{\mathrm e}^{5}-x^{2}}+5}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x-1)*exp(5)+2*x^2+2*x+1)*exp(x*exp(5)-x^2)+5)/(exp(x*exp(5)-x^2)^2+10*exp(x*exp(5)-x^2)+25),x,method=_
RETURNVERBOSE)

[Out]

(x+1)/(5+exp(x*(exp(5)-x)))

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maxima [A]  time = 0.50, size = 22, normalized size = 1.05 \begin {gather*} \frac {{\left (x + 1\right )} e^{\left (x^{2}\right )}}{5 \, e^{\left (x^{2}\right )} + e^{\left (x e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(5)+2*x^2+2*x+1)*exp(x*exp(5)-x^2)+5)/(exp(x*exp(5)-x^2)^2+10*exp(x*exp(5)-x^2)+25),x, a
lgorithm="maxima")

[Out]

(x + 1)*e^(x^2)/(5*e^(x^2) + e^(x*e^5))

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mupad [B]  time = 0.33, size = 19, normalized size = 0.90 \begin {gather*} \frac {x+1}{{\mathrm {e}}^{x\,{\mathrm {e}}^5-x^2}+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*exp(5) - x^2)*(2*x - exp(5)*(x + 1) + 2*x^2 + 1) + 5)/(10*exp(x*exp(5) - x^2) + exp(2*x*exp(5) - 2*
x^2) + 25),x)

[Out]

(x + 1)/(exp(x*exp(5) - x^2) + 5)

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sympy [A]  time = 0.14, size = 14, normalized size = 0.67 \begin {gather*} \frac {x + 1}{e^{- x^{2} + x e^{5}} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(5)+2*x**2+2*x+1)*exp(x*exp(5)-x**2)+5)/(exp(x*exp(5)-x**2)**2+10*exp(x*exp(5)-x**2)+25)
,x)

[Out]

(x + 1)/(exp(-x**2 + x*exp(5)) + 5)

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