3.60.57 \(\int \frac {1+2 x+e^{16-8 e^{x^2}+e^{2 x^2}} (-x+16 e^{x^2} x^3-4 e^{2 x^2} x^3)}{x} \, dx\) [5957]

Optimal. Leaf size=23 \[ 1+2 x-e^{\left (4-e^{x^2}\right )^2} x+\log (x) \]

[Out]

1+2*x+ln(x)-x*exp((4-exp(x^2))^2)

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Rubi [F]
time = 0.19, 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 {1+2 x+e^{16-8 e^{x^2}+e^{2 x^2}} \left (-x+16 e^{x^2} x^3-4 e^{2 x^2} x^3\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*x + E^(16 - 8*E^x^2 + E^(2*x^2))*(-x + 16*E^x^2*x^3 - 4*E^(2*x^2)*x^3))/x,x]

[Out]

2*x + Log[x] - Defer[Int][E^(-4 + E^x^2)^2, x] + 16*Defer[Int][E^(16 - 8*E^x^2 + E^(2*x^2) + x^2)*x^2, x] - 4*
Defer[Int][E^(16 - 8*E^x^2 + E^(2*x^2) + 2*x^2)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (16 e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2-4 e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2-\frac {-1-2 x+e^{\left (-4+e^{x^2}\right )^2} x}{x}\right ) \, dx\\ &=-\left (4 \int e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2 \, dx\right )+16 \int e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2 \, dx-\int \frac {-1-2 x+e^{\left (-4+e^{x^2}\right )^2} x}{x} \, dx\\ &=-\left (4 \int e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2 \, dx\right )+16 \int e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2 \, dx-\int \left (e^{\left (-4+e^{x^2}\right )^2}+\frac {-1-2 x}{x}\right ) \, dx\\ &=-\left (4 \int e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2 \, dx\right )+16 \int e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2 \, dx-\int e^{\left (-4+e^{x^2}\right )^2} \, dx-\int \frac {-1-2 x}{x} \, dx\\ &=-\left (4 \int e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2 \, dx\right )+16 \int e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2 \, dx-\int e^{\left (-4+e^{x^2}\right )^2} \, dx-\int \left (-2-\frac {1}{x}\right ) \, dx\\ &=2 x+\log (x)-4 \int e^{16-8 e^{x^2}+e^{2 x^2}+2 x^2} x^2 \, dx+16 \int e^{16-8 e^{x^2}+e^{2 x^2}+x^2} x^2 \, dx-\int e^{\left (-4+e^{x^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.15, size = 19, normalized size = 0.83 \begin {gather*} -\left (\left (-2+e^{\left (-4+e^{x^2}\right )^2}\right ) x\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + E^(16 - 8*E^x^2 + E^(2*x^2))*(-x + 16*E^x^2*x^3 - 4*E^(2*x^2)*x^3))/x,x]

[Out]

-((-2 + E^(-4 + E^x^2)^2)*x) + Log[x]

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Maple [A]
time = 6.37, size = 25, normalized size = 1.09

method	result	size
norman	\(2 x -{\mathrm e}^{{\mathrm e}^{2 x^{2}}-8 \,{\mathrm e}^{x^{2}}+16} x +\ln \left (x \right )\)	\(25\)
risch	\(2 x -{\mathrm e}^{{\mathrm e}^{2 x^{2}}-8 \,{\mathrm e}^{x^{2}}+16} x +\ln \left (x \right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3*exp(x^2)^2+16*x^3*exp(x^2)-x)*exp(exp(x^2)^2-8*exp(x^2)+16)+2*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

2*x-exp(exp(x^2)^2-8*exp(x^2)+16)*x+ln(x)

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Maxima [A]
time = 0.40, size = 24, normalized size = 1.04 \begin {gather*} -x e^{\left (e^{\left (2 \, x^{2}\right )} - 8 \, e^{\left (x^{2}\right )} + 16\right )} + 2 \, x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(x^2)^2+16*x^3*exp(x^2)-x)*exp(exp(x^2)^2-8*exp(x^2)+16)+2*x+1)/x,x, algorithm="maxima")

[Out]

-x*e^(e^(2*x^2) - 8*e^(x^2) + 16) + 2*x + log(x)

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Fricas [A]
time = 0.37, size = 24, normalized size = 1.04 \begin {gather*} -x e^{\left (e^{\left (2 \, x^{2}\right )} - 8 \, e^{\left (x^{2}\right )} + 16\right )} + 2 \, x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(x^2)^2+16*x^3*exp(x^2)-x)*exp(exp(x^2)^2-8*exp(x^2)+16)+2*x+1)/x,x, algorithm="fricas")

[Out]

-x*e^(e^(2*x^2) - 8*e^(x^2) + 16) + 2*x + log(x)

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Sympy [A]
time = 9.35, size = 24, normalized size = 1.04 \begin {gather*} - x e^{e^{2 x^{2}} - 8 e^{x^{2}} + 16} + 2 x + \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3*exp(x**2)**2+16*x**3*exp(x**2)-x)*exp(exp(x**2)**2-8*exp(x**2)+16)+2*x+1)/x,x)

[Out]

-x*exp(exp(2*x**2) - 8*exp(x**2) + 16) + 2*x + log(x)

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Giac [A]
time = 0.41, size = 24, normalized size = 1.04 \begin {gather*} -x e^{\left (e^{\left (2 \, x^{2}\right )} - 8 \, e^{\left (x^{2}\right )} + 16\right )} + 2 \, x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(x^2)^2+16*x^3*exp(x^2)-x)*exp(exp(x^2)^2-8*exp(x^2)+16)+2*x+1)/x,x, algorithm="giac")

[Out]

-x*e^(e^(2*x^2) - 8*e^(x^2) + 16) + 2*x + log(x)

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Mupad [B]
time = 0.13, size = 25, normalized size = 1.09 \begin {gather*} 2\,x+\ln \left (x\right )-x\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x^2}}\,{\mathrm {e}}^{-8\,{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - exp(exp(2*x^2) - 8*exp(x^2) + 16)*(x - 16*x^3*exp(x^2) + 4*x^3*exp(2*x^2)) + 1)/x,x)

[Out]

2*x + log(x) - x*exp(exp(2*x^2))*exp(-8*exp(x^2))*exp(16)

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