3.83.65 \(\int \frac {e^{-\frac {-4+e^5 x-2 x^2+x (i \pi +\log (2-\sqrt [4]{e}))}{x}} (-16-4 x+8 x^2)}{x^3} \, dx\)

Optimal. Leaf size=37 \[ \frac {4 e^{-e^5-i \pi +\frac {4}{x}+2 x}}{\left (2-\sqrt [4]{e}\right ) x} \]

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Rubi [B]  time = 0.16, antiderivative size = 115, normalized size of antiderivative = 3.11, number of steps used = 1, number of rules used = 1, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2288} \begin {gather*} \frac {8 \left (2-x^2\right ) e^{\frac {2 x^2-e^5 x-x \left (\log \left (2-\sqrt [4]{e}\right )+i \pi \right )+4}{x}}}{x^3 \left (\frac {2 x^2-e^5 x-x \left (\log \left (2-\sqrt [4]{e}\right )+i \pi \right )+4}{x^2}+\frac {-4 x+i \pi +e^5+\log \left (2-\sqrt [4]{e}\right )}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 - 4*x + 8*x^2)/(E^((-4 + E^5*x - 2*x^2 + x*(I*Pi + Log[2 - E^(1/4)]))/x)*x^3),x]

[Out]

(8*E^((4 - E^5*x + 2*x^2 - x*(I*Pi + Log[2 - E^(1/4)]))/x)*(2 - x^2))/(x^3*((E^5 + I*Pi - 4*x + Log[2 - E^(1/4
)])/x + (4 - E^5*x + 2*x^2 - x*(I*Pi + Log[2 - E^(1/4)]))/x^2))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {8 e^{\frac {4-e^5 x+2 x^2-x \left (i \pi +\log \left (2-\sqrt [4]{e}\right )\right )}{x}} \left (2-x^2\right )}{x^3 \left (\frac {e^5+i \pi -4 x+\log \left (2-\sqrt [4]{e}\right )}{x}+\frac {4-e^5 x+2 x^2-x \left (i \pi +\log \left (2-\sqrt [4]{e}\right )\right )}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.81 \begin {gather*} \frac {4 e^{-e^5+\frac {4}{x}+2 x}}{\left (-2+\sqrt [4]{e}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 4*x + 8*x^2)/(E^((-4 + E^5*x - 2*x^2 + x*(I*Pi + Log[2 - E^(1/4)]))/x)*x^3),x]

[Out]

(4*E^(-E^5 + 4/x + 2*x))/((-2 + E^(1/4))*x)

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fricas [A]  time = 1.07, size = 30, normalized size = 0.81 \begin {gather*} \frac {4 \, e^{\left (\frac {2 \, x^{2} - x e^{5} - x \log \left (e^{\frac {1}{4}} - 2\right ) + 4}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-4*x-16)/x^3/exp((x*log(exp(1/4)-2)+x*exp(5)-2*x^2-4)/x),x, algorithm="fricas")

[Out]

4*e^((2*x^2 - x*e^5 - x*log(e^(1/4) - 2) + 4)/x)/x

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giac [A]  time = 0.16, size = 29, normalized size = 0.78 \begin {gather*} \frac {4 \, e^{\left (\frac {2 \, x^{2} - x e^{5} + 4}{x}\right )}}{x e^{\frac {1}{4}} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-4*x-16)/x^3/exp((x*log(exp(1/4)-2)+x*exp(5)-2*x^2-4)/x),x, algorithm="giac")

[Out]

4*e^((2*x^2 - x*e^5 + 4)/x)/(x*e^(1/4) - 2*x)

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maple [A]  time = 0.29, size = 29, normalized size = 0.78

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{-\frac {x \,{\mathrm e}^{5}-2 x^{2}-4}{x}}}{x \left ({\mathrm e}^{\frac {1}{4}}-2\right )}\)	\(29\)
gosper	\(\frac {4 \,{\mathrm e}^{-\frac {x \ln \left ({\mathrm e}^{\frac {1}{4}}-2\right )+x \,{\mathrm e}^{5}-2 x^{2}-4}{x}}}{x}\)	\(31\)
norman	\(\frac {4 \,{\mathrm e}^{-\frac {x \ln \left ({\mathrm e}^{\frac {1}{4}}-2\right )+x \,{\mathrm e}^{5}-2 x^{2}-4}{x}}}{x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2-4*x-16)/x^3/exp((x*ln(exp(1/4)-2)+x*exp(5)-2*x^2-4)/x),x,method=_RETURNVERBOSE)

[Out]

4/x/(exp(1/4)-2)*exp(-(x*exp(5)-2*x^2-4)/x)

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maxima [A]  time = 0.48, size = 28, normalized size = 0.76 \begin {gather*} \frac {4 \, e^{\left (2 \, x + \frac {4}{x}\right )}}{x {\left (e^{\left (e^{5} + \frac {1}{4}\right )} - 2 \, e^{\left (e^{5}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-4*x-16)/x^3/exp((x*log(exp(1/4)-2)+x*exp(5)-2*x^2-4)/x),x, algorithm="maxima")

[Out]

4*e^(2*x + 4/x)/(x*(e^(e^5 + 1/4) - 2*e^(e^5)))

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mupad [B]  time = 4.91, size = 26, normalized size = 0.70 \begin {gather*} \frac {4\,{\mathrm {e}}^{-{\mathrm {e}}^5}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{4/x}}{x\,\left ({\mathrm {e}}^{1/4}-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(x*log(exp(1/4) - 2) + x*exp(5) - 2*x^2 - 4)/x)*(4*x - 8*x^2 + 16))/x^3,x)

[Out]

(4*exp(-exp(5))*exp(2*x)*exp(4/x))/(x*(exp(1/4) - 2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**2-4*x-16)/x**3/exp((x*ln(exp(1/4)-2)+x*exp(5)-2*x**2-4)/x),x)

[Out]

Timed out

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