3.12.88 \(\int \frac {-3 x+2 x^2+4 e^{-2 x+4 e^{-2 x} x^2} (2-2 x) x^2}{x} \, dx\)

Optimal. Leaf size=20 \[ -3+e^{4 e^{-2 x} x^2}-3 x+x^2 \]

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Rubi [F]  time = 0.39, 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 {-3 x+2 x^2+4 e^{-2 x+4 e^{-2 x} x^2} (2-2 x) x^2}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3+2 x-8 e^{2 x \left (-1+2 e^{-2 x} x\right )} (-1+x) x\right ) \, dx\\ &=-3 x+x^2-8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} (-1+x) x \, dx\\ &=-3 x+x^2-8 \int \left (-e^{2 x \left (-1+2 e^{-2 x} x\right )} x+e^{2 x \left (-1+2 e^{-2 x} x\right )} x^2\right ) \, dx\\ &=-3 x+x^2+8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} x \, dx-8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 19, normalized size = 0.95 \begin {gather*} e^{4 e^{-2 x} x^2}-3 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^((4*x^2)/E^(2*x)) - 3*x + x^2

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fricas [B]  time = 0.88, size = 56, normalized size = 2.80 \begin {gather*} {\left ({\left (x^{2} - 3 \, x\right )} e^{\left (-2 \, x + \log \left (4 \, x^{2}\right )\right )} + e^{\left (-2 \, x + e^{\left (-2 \, x + \log \left (4 \, x^{2}\right )\right )} + \log \left (4 \, x^{2}\right )\right )}\right )} e^{\left (2 \, x - \log \left (4 \, x^{2}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2 - 3*x)*e^(-2*x + log(4*x^2)) + e^(-2*x + e^(-2*x + log(4*x^2)) + log(4*x^2)))*e^(2*x - log(4*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} - 2 \, {\left (x - 1\right )} e^{\left (-2 \, x + e^{\left (-2 \, x + \log \left (4 \, x^{2}\right )\right )} + \log \left (4 \, x^{2}\right )\right )} - 3 \, x}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^2 - 2*(x - 1)*e^(-2*x + e^(-2*x + log(4*x^2)) + log(4*x^2)) - 3*x)/x, x)

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maple [A]  time = 0.08, size = 18, normalized size = 0.90

method	result	size
risch	\(x^{2}-3 x +{\mathrm e}^{4 x^{2} {\mathrm e}^{-2 x}}\)	\(18\)
default	\(x^{2}-3 x +{\mathrm e}^{{\mathrm e}^{\ln \left (4 x^{2}\right )-2 x}}\)	\(20\)
norman	\(x^{2}-3 x +{\mathrm e}^{{\mathrm e}^{\ln \left (4 x^{2}\right )-2 x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2-3*x+exp(4*x^2*exp(-2*x))

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maxima [A]  time = 0.50, size = 17, normalized size = 0.85 \begin {gather*} x^{2} - 3 \, x + e^{\left (4 \, x^{2} e^{\left (-2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - 3*x + e^(4*x^2*e^(-2*x))

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mupad [B]  time = 0.90, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{-2\,x}}-3\,x+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - 2*x^2 + exp(log(4*x^2) - 2*x)*exp(exp(log(4*x^2) - 2*x))*(2*x - 2))/x,x)

[Out]

exp(4*x^2*exp(-2*x)) - 3*x + x^2

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sympy [A]  time = 0.20, size = 17, normalized size = 0.85 \begin {gather*} x^{2} - 3 x + e^{4 x^{2} e^{- 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*exp(ln(4*x**2)-2*x)*exp(exp(ln(4*x**2)-2*x))+2*x**2-3*x)/x,x)

[Out]

x**2 - 3*x + exp(4*x**2*exp(-2*x))

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