3.62.57 \(\int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} (1+2 x+e^{x^2} (10 x-2 x^2-2 x^3))}{-5+x+x^2}}{-5+x+x^2} \, dx\)

Optimal. Leaf size=24 \[ -e^{e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}}+x \]

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Rubi [F]  time = 25.21, 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+x+x^2+\frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + x + x^2 + (E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5 + x + x^2))*(1 + 2*x + E^x^2*(10*x -
2*x^2 - 2*x^3)))/(-5 + x + x^2))/(-5 + x + x^2),x]

[Out]

x + (4*Defer[Int][E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5 + x + x^2))/(-1 + Sqrt[21] - 2*x)^2, x])
/21 - (4*(1 - Sqrt[21])*Defer[Int][E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5 + x + x^2))/(-1 + Sqrt[
21] - 2*x)^2, x])/21 - (2*(21 - Sqrt[21])*Defer[Int][E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + x^2 + E^E^x^2/(-5
+ x + x^2))/(1 - Sqrt[21] + 2*x), x])/21 + (4*Defer[Int][E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5
+ x + x^2))/(1 + Sqrt[21] + 2*x)^2, x])/21 - (4*(1 + Sqrt[21])*Defer[Int][E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)
) + E^E^x^2/(-5 + x + x^2))/(1 + Sqrt[21] + 2*x)^2, x])/21 - (2*(21 + Sqrt[21])*Defer[Int][E^(E^x^2 + E^(E^E^x
^2/(-5 + x + x^2)) + x^2 + E^E^x^2/(-5 + x + x^2))/(1 + Sqrt[21] + 2*x), x])/21

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) \left (-1-2 x-10 e^{x^2} x+2 e^{x^2} x^2+2 e^{x^2} x^3\right )}{\left (-5+x+x^2\right )^2}\right ) \, dx\\ &=x-\int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) \left (-1-2 x-10 e^{x^2} x+2 e^{x^2} x^2+2 e^{x^2} x^3\right )}{\left (-5+x+x^2\right )^2} \, dx\\ &=x-\int \left (\frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) (-1-2 x)}{\left (-5+x+x^2\right )^2}+\frac {2 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) x}{-5+x+x^2}\right ) \, dx\\ &=x-2 \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) x}{-5+x+x^2} \, dx-\int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) (-1-2 x)}{\left (-5+x+x^2\right )^2} \, dx\\ &=x-2 \int \left (\frac {\left (1-\frac {1}{\sqrt {21}}\right ) \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1-\sqrt {21}+2 x}+\frac {\left (1+\frac {1}{\sqrt {21}}\right ) \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1+\sqrt {21}+2 x}\right ) \, dx-\int \left (-\frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (-5+x+x^2\right )^2}-\frac {2 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) x}{\left (-5+x+x^2\right )^2}\right ) \, dx\\ &=x+2 \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right ) x}{\left (-5+x+x^2\right )^2} \, dx-\frac {1}{21} \left (2 \left (21-\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1-\sqrt {21}+2 x} \, dx-\frac {1}{21} \left (2 \left (21+\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1+\sqrt {21}+2 x} \, dx+\int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (-5+x+x^2\right )^2} \, dx\\ &=x+2 \int \left (\frac {2 \left (-1+\sqrt {21}\right ) \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \left (-1+\sqrt {21}-2 x\right )^2}-\frac {2 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \sqrt {21} \left (-1+\sqrt {21}-2 x\right )}+\frac {2 \left (-1-\sqrt {21}\right ) \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \left (1+\sqrt {21}+2 x\right )^2}-\frac {2 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \sqrt {21} \left (1+\sqrt {21}+2 x\right )}\right ) \, dx-\frac {1}{21} \left (2 \left (21-\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1-\sqrt {21}+2 x} \, dx-\frac {1}{21} \left (2 \left (21+\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1+\sqrt {21}+2 x} \, dx+\int \left (\frac {4 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \left (-1+\sqrt {21}-2 x\right )^2}+\frac {4 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \sqrt {21} \left (-1+\sqrt {21}-2 x\right )}+\frac {4 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \left (1+\sqrt {21}+2 x\right )^2}+\frac {4 \exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{21 \sqrt {21} \left (1+\sqrt {21}+2 x\right )}\right ) \, dx\\ &=x+\frac {4}{21} \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (-1+\sqrt {21}-2 x\right )^2} \, dx+\frac {4}{21} \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (1+\sqrt {21}+2 x\right )^2} \, dx-\frac {1}{21} \left (4 \left (1-\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (-1+\sqrt {21}-2 x\right )^2} \, dx-\frac {1}{21} \left (2 \left (21-\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1-\sqrt {21}+2 x} \, dx-\frac {1}{21} \left (4 \left (1+\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{\left (1+\sqrt {21}+2 x\right )^2} \, dx-\frac {1}{21} \left (2 \left (21+\sqrt {21}\right )\right ) \int \frac {\exp \left (e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+x^2+\frac {e^{e^{x^2}}}{-5+x+x^2}\right )}{1+\sqrt {21}+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 24, normalized size = 1.00 \begin {gather*} -e^{e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + x + x^2 + (E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5 + x + x^2))*(1 + 2*x + E^x^2*(1
0*x - 2*x^2 - 2*x^3)))/(-5 + x + x^2))/(-5 + x + x^2),x]

[Out]

-E^E^(E^E^x^2/(-5 + x + x^2)) + x

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fricas [B]  time = 0.50, size = 114, normalized size = 4.75 \begin {gather*} {\left (x e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} - \log \left (x^{2} + x - 5\right )\right )} - e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + e^{\left (e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} e^{\left (-e^{\left (x^{2}\right )} - e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + \log \left (x^{2} + x - 5\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2))*exp(exp(-log(x^2+x-5)+exp(x^2)))*e
xp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x^2+x-5)/(x^2+x-5),x, algorithm="fricas")

[Out]

(x*e^(e^(x^2) + e^(e^(x^2) - log(x^2 + x - 5)) - log(x^2 + x - 5)) - e^(e^(x^2) + e^(e^(x^2) - log(x^2 + x - 5
)) + e^(e^(e^(x^2) - log(x^2 + x - 5))) - log(x^2 + x - 5)))*e^(-e^(x^2) - e^(e^(x^2) - log(x^2 + x - 5)) + lo
g(x^2 + x - 5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - {\left (2 \, {\left (x^{3} + x^{2} - 5 \, x\right )} e^{\left (x^{2}\right )} - 2 \, x - 1\right )} e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + e^{\left (e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} - \log \left (x^{2} + x - 5\right )\right )} + x - 5}{x^{2} + x - 5}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2))*exp(exp(-log(x^2+x-5)+exp(x^2)))*e
xp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x^2+x-5)/(x^2+x-5),x, algorithm="giac")

[Out]

integrate((x^2 - (2*(x^3 + x^2 - 5*x)*e^(x^2) - 2*x - 1)*e^(e^(x^2) + e^(e^(x^2) - log(x^2 + x - 5)) + e^(e^(e
^(x^2) - log(x^2 + x - 5))) - log(x^2 + x - 5)) + x - 5)/(x^2 + x - 5), x)

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maple [A]  time = 0.62, size = 21, normalized size = 0.88

method	result	size
risch	\(x -{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x^{2}}}}{x^{2}+x -5}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-ln(x^2+x-5)+exp(x^2))*exp(exp(-ln(x^2+x-5)+exp(x^2)))*exp(exp(e
xp(-ln(x^2+x-5)+exp(x^2))))+x^2+x-5)/(x^2+x-5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.78, size = 20, normalized size = 0.83 \begin {gather*} x - e^{\left (e^{\left (\frac {e^{\left (e^{\left (x^{2}\right )}\right )}}{x^{2} + x - 5}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2))*exp(exp(-log(x^2+x-5)+exp(x^2)))*e
xp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x^2+x-5)/(x^2+x-5),x, algorithm="maxima")

[Out]

x - e^(e^(e^(e^(x^2))/(x^2 + x - 5)))

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mupad [B]  time = 4.20, size = 20, normalized size = 0.83 \begin {gather*} x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^2}}}{x^2+x-5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^2 + exp(exp(exp(exp(x^2) - log(x + x^2 - 5))))*exp(exp(x^2) - log(x + x^2 - 5))*exp(exp(exp(x^2) -
log(x + x^2 - 5)))*(2*x - exp(x^2)*(2*x^2 - 10*x + 2*x^3) + 1) - 5)/(x + x^2 - 5),x)

[Out]

x - exp(exp(exp(exp(x^2))/(x + x^2 - 5)))

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sympy [A]  time = 60.51, size = 17, normalized size = 0.71 \begin {gather*} x - e^{e^{\frac {e^{e^{x^{2}}}}{x^{2} + x - 5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**3-2*x**2+10*x)*exp(x**2)+2*x+1)*exp(-ln(x**2+x-5)+exp(x**2))*exp(exp(-ln(x**2+x-5)+exp(x**2
)))*exp(exp(exp(-ln(x**2+x-5)+exp(x**2))))+x**2+x-5)/(x**2+x-5),x)

[Out]

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

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