3.38.23 \(\int \frac {e^{4+4 e^{-6+x} x+e^{-12+2 x} x^2} (-1+e^{-6+x} (16 x+16 x^2)+e^{-12+2 x} (8 x^2+8 x^3)+(e^{-6+x} (-4 x-4 x^2)+e^{-12+2 x} (-2 x^2-2 x^3)) \log (x))}{x} \, dx\)

Optimal. Leaf size=22 \[ -4+e^{\left (2+e^{-6+x} x\right )^2} (4-\log (x)) \]

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Rubi [B]  time = 0.68, antiderivative size = 126, normalized size of antiderivative = 5.73, number of steps used = 1, number of rules used = 1, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2288} \begin {gather*} \frac {e^{e^{2 x-12} x^2+4 e^{x-6} x+4} \left (8 e^{x-6} \left (x^2+x\right )+4 e^{2 x-12} \left (x^3+x^2\right )-\left (2 e^{x-6} \left (x^2+x\right )+e^{2 x-12} \left (x^3+x^2\right )\right ) \log (x)\right )}{x \left (e^{2 x-12} x^2+2 e^{x-6} x+e^{2 x-12} x+2 e^{x-6}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4 + 4*E^(-6 + x)*x + E^(-12 + 2*x)*x^2)*(-1 + E^(-6 + x)*(16*x + 16*x^2) + E^(-12 + 2*x)*(8*x^2 + 8*x^
3) + (E^(-6 + x)*(-4*x - 4*x^2) + E^(-12 + 2*x)*(-2*x^2 - 2*x^3))*Log[x]))/x,x]

[Out]

(E^(4 + 4*E^(-6 + x)*x + E^(-12 + 2*x)*x^2)*(8*E^(-6 + x)*(x + x^2) + 4*E^(-12 + 2*x)*(x^2 + x^3) - (2*E^(-6 +
x)*(x + x^2) + E^(-12 + 2*x)*(x^2 + x^3))*Log[x]))/(x*(2*E^(-6 + x) + 2*E^(-6 + x)*x + E^(-12 + 2*x)*x + E^(-
12 + 2*x)*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 {e^{4+4 e^{-6+x} x+e^{-12+2 x} x^2} \left (8 e^{-6+x} \left (x+x^2\right )+4 e^{-12+2 x} \left (x^2+x^3\right )-\left (2 e^{-6+x} \left (x+x^2\right )+e^{-12+2 x} \left (x^2+x^3\right )\right ) \log (x)\right )}{x \left (2 e^{-6+x}+2 e^{-6+x} x+e^{-12+2 x} x+e^{-12+2 x} x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 25, normalized size = 1.14 \begin {gather*} -e^{\frac {\left (2 e^6+e^x x\right )^2}{e^{12}}} (-4+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + 4*E^(-6 + x)*x + E^(-12 + 2*x)*x^2)*(-1 + E^(-6 + x)*(16*x + 16*x^2) + E^(-12 + 2*x)*(8*x^2
+ 8*x^3) + (E^(-6 + x)*(-4*x - 4*x^2) + E^(-12 + 2*x)*(-2*x^2 - 2*x^3))*Log[x]))/x,x]

[Out]

-(E^((2*E^6 + E^x*x)^2/E^12)*(-4 + Log[x]))

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fricas [A]  time = 0.72, size = 26, normalized size = 1.18 \begin {gather*} -{\left (\log \relax (x) - 4\right )} e^{\left (x^{2} e^{\left (2 \, x - 12\right )} + 4 \, x e^{\left (x - 6\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2)*exp(x-6)^2+(-4*x^2-4*x)*exp(x-6))*log(x)+(8*x^3+8*x^2)*exp(x-6)^2+(16*x^2+16*x)*exp
(x-6)-1)*exp(x^2*exp(x-6)^2+4*x*exp(x-6)+4)/x,x, algorithm="fricas")

[Out]

-(log(x) - 4)*e^(x^2*e^(2*x - 12) + 4*x*e^(x - 6) + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (8 \, {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x - 12\right )} + 16 \, {\left (x^{2} + x\right )} e^{\left (x - 6\right )} - 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (2 \, x - 12\right )} + 2 \, {\left (x^{2} + x\right )} e^{\left (x - 6\right )}\right )} \log \relax (x) - 1\right )} e^{\left (x^{2} e^{\left (2 \, x - 12\right )} + 4 \, x e^{\left (x - 6\right )} + 4\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2)*exp(x-6)^2+(-4*x^2-4*x)*exp(x-6))*log(x)+(8*x^3+8*x^2)*exp(x-6)^2+(16*x^2+16*x)*exp
(x-6)-1)*exp(x^2*exp(x-6)^2+4*x*exp(x-6)+4)/x,x, algorithm="giac")

[Out]

integrate((8*(x^3 + x^2)*e^(2*x - 12) + 16*(x^2 + x)*e^(x - 6) - 2*((x^3 + x^2)*e^(2*x - 12) + 2*(x^2 + x)*e^(
x - 6))*log(x) - 1)*e^(x^2*e^(2*x - 12) + 4*x*e^(x - 6) + 4)/x, x)

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maple [A]  time = 0.08, size = 28, normalized size = 1.27

method	result	size
risch	\(\left (-\ln \relax (x )+4\right ) {\mathrm e}^{x^{2} {\mathrm e}^{2 x -12}+4 x \,{\mathrm e}^{x -6}+4}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^3-2*x^2)*exp(x-6)^2+(-4*x^2-4*x)*exp(x-6))*ln(x)+(8*x^3+8*x^2)*exp(x-6)^2+(16*x^2+16*x)*exp(x-6)-1
)*exp(x^2*exp(x-6)^2+4*x*exp(x-6)+4)/x,x,method=_RETURNVERBOSE)

[Out]

(-ln(x)+4)*exp(x^2*exp(2*x-12)+4*x*exp(x-6)+4)

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maxima [A]  time = 0.54, size = 31, normalized size = 1.41 \begin {gather*} -{\left (e^{4} \log \relax (x) - 4 \, e^{4}\right )} e^{\left (x^{2} e^{\left (2 \, x - 12\right )} + 4 \, x e^{\left (x - 6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3-2*x^2)*exp(x-6)^2+(-4*x^2-4*x)*exp(x-6))*log(x)+(8*x^3+8*x^2)*exp(x-6)^2+(16*x^2+16*x)*exp
(x-6)-1)*exp(x^2*exp(x-6)^2+4*x*exp(x-6)+4)/x,x, algorithm="maxima")

[Out]

-(e^4*log(x) - 4*e^4)*e^(x^2*e^(2*x - 12) + 4*x*e^(x - 6))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x-6}+x^2\,{\mathrm {e}}^{2\,x-12}+4}\,\left ({\mathrm {e}}^{x-6}\,\left (16\,x^2+16\,x\right )-\ln \relax (x)\,\left ({\mathrm {e}}^{x-6}\,\left (4\,x^2+4\,x\right )+{\mathrm {e}}^{2\,x-12}\,\left (2\,x^3+2\,x^2\right )\right )+{\mathrm {e}}^{2\,x-12}\,\left (8\,x^3+8\,x^2\right )-1\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x*exp(x - 6) + x^2*exp(2*x - 12) + 4)*(exp(x - 6)*(16*x + 16*x^2) - log(x)*(exp(x - 6)*(4*x + 4*x^2
) + exp(2*x - 12)*(2*x^2 + 2*x^3)) + exp(2*x - 12)*(8*x^2 + 8*x^3) - 1))/x,x)

[Out]

int((exp(4*x*exp(x - 6) + x^2*exp(2*x - 12) + 4)*(exp(x - 6)*(16*x + 16*x^2) - log(x)*(exp(x - 6)*(4*x + 4*x^2
) + exp(2*x - 12)*(2*x^2 + 2*x^3)) + exp(2*x - 12)*(8*x^2 + 8*x^3) - 1))/x, x)

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sympy [A]  time = 0.65, size = 26, normalized size = 1.18 \begin {gather*} \left (4 - \log {\relax (x )}\right ) e^{x^{2} e^{2 x - 12} + 4 x e^{x - 6} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**3-2*x**2)*exp(x-6)**2+(-4*x**2-4*x)*exp(x-6))*ln(x)+(8*x**3+8*x**2)*exp(x-6)**2+(16*x**2+16
*x)*exp(x-6)-1)*exp(x**2*exp(x-6)**2+4*x*exp(x-6)+4)/x,x)

[Out]

(4 - log(x))*exp(x**2*exp(2*x - 12) + 4*x*exp(x - 6) + 4)

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