3.59.5 \(\int \frac {e^{e^{180-2 x} x^8} (-1+e^{180-2 x} (8-2 x) x^8)}{x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac {e^{e^{2 (78-x+4 (3+\log (x)))}}}{x} \]

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Rubi [B]  time = 0.08, antiderivative size = 54, normalized size of antiderivative = 2.57, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2288} \begin {gather*} \frac {e^{e^{180-2 x} x^8-2 x+180} (4-x) x^6}{4 e^{180-2 x} x^7-e^{180-2 x} x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^(180 - 2*x)*x^8)*(-1 + E^(180 - 2*x)*(8 - 2*x)*x^8))/x^2,x]

[Out]

(E^(180 - 2*x + E^(180 - 2*x)*x^8)*(4 - x)*x^6)/(4*E^(180 - 2*x)*x^7 - E^(180 - 2*x)*x^8)

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 {e^{180-2 x+e^{180-2 x} x^8} (4-x) x^6}{4 e^{180-2 x} x^7-e^{180-2 x} x^8}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 17, normalized size = 0.81 \begin {gather*} \frac {e^{e^{180-2 x} x^8}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^(180 - 2*x)*x^8)*(-1 + E^(180 - 2*x)*(8 - 2*x)*x^8))/x^2,x]

[Out]

E^(E^(180 - 2*x)*x^8)/x

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fricas [A]  time = 0.52, size = 15, normalized size = 0.71 \begin {gather*} \frac {e^{\left (e^{\left (-2 \, x + 8 \, \log \relax (x) + 180\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(8*log(x)-2*x+180)-1)*exp(exp(8*log(x)-2*x+180))/x^2,x, algorithm="fricas")

[Out]

e^(e^(-2*x + 8*log(x) + 180))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, {\left (x - 4\right )} e^{\left (-2 \, x + 8 \, \log \relax (x) + 180\right )} + 1\right )} e^{\left (e^{\left (-2 \, x + 8 \, \log \relax (x) + 180\right )}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(8*log(x)-2*x+180)-1)*exp(exp(8*log(x)-2*x+180))/x^2,x, algorithm="giac")

[Out]

integrate(-(2*(x - 4)*e^(-2*x + 8*log(x) + 180) + 1)*e^(e^(-2*x + 8*log(x) + 180))/x^2, x)

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maple [A]  time = 0.09, size = 16, normalized size = 0.76

method	result	size
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{8 \ln \relax (x )-2 x +180}}}{x}\)	\(16\)
risch	\(\frac {{\mathrm e}^{x^{8} {\mathrm e}^{180-2 x}}}{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+8)*exp(8*ln(x)-2*x+180)-1)*exp(exp(8*ln(x)-2*x+180))/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(exp(8*ln(x)-2*x+180))/x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (2 \, {\left (x - 4\right )} x^{8} e^{\left (-2 \, x + 180\right )} + 1\right )} e^{\left (x^{8} e^{\left (-2 \, x + 180\right )}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(8*log(x)-2*x+180)-1)*exp(exp(8*log(x)-2*x+180))/x^2,x, algorithm="maxima")

[Out]

-integrate((2*(x - 4)*x^8*e^(-2*x + 180) + 1)*e^(x^8*e^(-2*x + 180))/x^2, x)

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mupad [B]  time = 4.17, size = 15, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{x^8\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{180}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(8*log(x) - 2*x + 180))*(exp(8*log(x) - 2*x + 180)*(2*x - 8) + 1))/x^2,x)

[Out]

exp(x^8*exp(-2*x)*exp(180))/x

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sympy [A]  time = 0.15, size = 12, normalized size = 0.57 \begin {gather*} \frac {e^{x^{8} e^{180 - 2 x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(8*ln(x)-2*x+180)-1)*exp(exp(8*ln(x)-2*x+180))/x**2,x)

[Out]

exp(x**8*exp(180 - 2*x))/x

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