3.97.25 \(\int e^{-9 x-24 x^3-16 x^5} (40+e^{9 x+24 x^3+16 x^5} (-3-x)+(40+e^{9 x+24 x^3+16 x^5} (-3-2 x)-360 x-2880 x^3-3200 x^5) \log (x)) \, dx\)

Optimal. Leaf size=27 \[ \left (40 e^{-x \left (3+4 x^2\right )^2} x-x (3+x)\right ) \log (x) \]

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Rubi [B]  time = 3.29, antiderivative size = 69, normalized size of antiderivative = 2.56, number of steps used = 5, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6741, 6742, 6688, 2288, 2554} \begin {gather*} x^2 (-\log (x))+\frac {40 e^{-x \left (4 x^2+3\right )^2} \left (80 x^5+72 x^3+9 x\right ) \log (x)}{16 \left (4 x^2+3\right ) x^2+\left (4 x^2+3\right )^2}-3 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-9*x - 24*x^3 - 16*x^5)*(40 + E^(9*x + 24*x^3 + 16*x^5)*(-3 - x) + (40 + E^(9*x + 24*x^3 + 16*x^5)*(-3
- 2*x) - 360*x - 2880*x^3 - 3200*x^5)*Log[x]),x]

[Out]

-3*x*Log[x] - x^2*Log[x] + (40*(9*x + 72*x^3 + 80*x^5)*Log[x])/(E^(x*(3 + 4*x^2)^2)*(16*x^2*(3 + 4*x^2) + (3 +
4*x^2)^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]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-x \left (3+4 x^2\right )^2} \left (40+e^{9 x+24 x^3+16 x^5} (-3-x)+\left (40+e^{9 x+24 x^3+16 x^5} (-3-2 x)-360 x-2880 x^3-3200 x^5\right ) \log (x)\right ) \, dx\\ &=\int \left (-3+40 e^{-x \left (3+4 x^2\right )^2}-x-e^{-x \left (3+4 x^2\right )^2} \left (-40+3 e^{x \left (3+4 x^2\right )^2}+360 x+2 e^{x \left (3+4 x^2\right )^2} x+2880 x^3+3200 x^5\right ) \log (x)\right ) \, dx\\ &=-3 x-\frac {x^2}{2}+40 \int e^{-x \left (3+4 x^2\right )^2} \, dx-\int e^{-x \left (3+4 x^2\right )^2} \left (-40+3 e^{x \left (3+4 x^2\right )^2}+360 x+2 e^{x \left (3+4 x^2\right )^2} x+2880 x^3+3200 x^5\right ) \log (x) \, dx\\ &=-3 x-\frac {x^2}{2}-3 x \log (x)-x^2 \log (x)+\frac {40 e^{-x \left (3+4 x^2\right )^2} \left (9 x+72 x^3+80 x^5\right ) \log (x)}{16 x^2 \left (3+4 x^2\right )+\left (3+4 x^2\right )^2}+40 \int e^{-x \left (3+4 x^2\right )^2} \, dx+\int \left (3-40 e^{-x \left (3+4 x^2\right )^2}+x\right ) \, dx\\ &=-3 x \log (x)-x^2 \log (x)+\frac {40 e^{-x \left (3+4 x^2\right )^2} \left (9 x+72 x^3+80 x^5\right ) \log (x)}{16 x^2 \left (3+4 x^2\right )+\left (3+4 x^2\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 25, normalized size = 0.93 \begin {gather*} \left (-3+40 e^{-x \left (3+4 x^2\right )^2}-x\right ) x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-9*x - 24*x^3 - 16*x^5)*(40 + E^(9*x + 24*x^3 + 16*x^5)*(-3 - x) + (40 + E^(9*x + 24*x^3 + 16*x^5
)*(-3 - 2*x) - 360*x - 2880*x^3 - 3200*x^5)*Log[x]),x]

[Out]

(-3 + 40/E^(x*(3 + 4*x^2)^2) - x)*x*Log[x]

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fricas [A]  time = 0.64, size = 46, normalized size = 1.70 \begin {gather*} -{\left ({\left (x^{2} + 3 \, x\right )} e^{\left (16 \, x^{5} + 24 \, x^{3} + 9 \, x\right )} - 40 \, x\right )} e^{\left (-16 \, x^{5} - 24 \, x^{3} - 9 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-3)*exp(16*x^5+24*x^3+9*x)-3200*x^5-2880*x^3-360*x+40)*log(x)+(-3-x)*exp(16*x^5+24*x^3+9*x)+4
0)/exp(16*x^5+24*x^3+9*x),x, algorithm="fricas")

[Out]

-((x^2 + 3*x)*e^(16*x^5 + 24*x^3 + 9*x) - 40*x)*e^(-16*x^5 - 24*x^3 - 9*x)*log(x)

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giac [A]  time = 0.18, size = 33, normalized size = 1.22 \begin {gather*} -x^{2} \log \relax (x) + 40 \, x e^{\left (-16 \, x^{5} - 24 \, x^{3} - 9 \, x\right )} \log \relax (x) - 3 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-3)*exp(16*x^5+24*x^3+9*x)-3200*x^5-2880*x^3-360*x+40)*log(x)+(-3-x)*exp(16*x^5+24*x^3+9*x)+4
0)/exp(16*x^5+24*x^3+9*x),x, algorithm="giac")

[Out]

-x^2*log(x) + 40*x*e^(-16*x^5 - 24*x^3 - 9*x)*log(x) - 3*x*log(x)

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maple [A]  time = 0.06, size = 49, normalized size = 1.81

method	result	size
risch	\(-x \left ({\mathrm e}^{\left (4 x^{2}+3\right )^{2} x} x +3 \,{\mathrm e}^{\left (4 x^{2}+3\right )^{2} x}-40\right ) {\mathrm e}^{-\left (4 x^{2}+3\right )^{2} x} \ln \relax (x )\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x-3)*exp(16*x^5+24*x^3+9*x)-3200*x^5-2880*x^3-360*x+40)*ln(x)+(-3-x)*exp(16*x^5+24*x^3+9*x)+40)/exp(
16*x^5+24*x^3+9*x),x,method=_RETURNVERBOSE)

[Out]

-x*(exp((4*x^2+3)^2*x)*x+3*exp((4*x^2+3)^2*x)-40)*exp(-(4*x^2+3)^2*x)*ln(x)

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maxima [B]  time = 0.42, size = 62, normalized size = 2.30 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (80 \, x e^{\left (-16 \, x^{5}\right )} \log \relax (x) + {\left (x^{2} - 2 \, {\left (x^{2} + 3 \, x\right )} \log \relax (x) + 6 \, x\right )} e^{\left (24 \, x^{3} + 9 \, x\right )}\right )} e^{\left (-24 \, x^{3} - 9 \, x\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-3)*exp(16*x^5+24*x^3+9*x)-3200*x^5-2880*x^3-360*x+40)*log(x)+(-3-x)*exp(16*x^5+24*x^3+9*x)+4
0)/exp(16*x^5+24*x^3+9*x),x, algorithm="maxima")

[Out]

-1/2*x^2 + 1/2*(80*x*e^(-16*x^5)*log(x) + (x^2 - 2*(x^2 + 3*x)*log(x) + 6*x)*e^(24*x^3 + 9*x))*e^(-24*x^3 - 9*
x) - 3*x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -{\mathrm {e}}^{-16\,x^5-24\,x^3-9\,x}\,\left ({\mathrm {e}}^{16\,x^5+24\,x^3+9\,x}\,\left (x+3\right )+\ln \relax (x)\,\left (360\,x+{\mathrm {e}}^{16\,x^5+24\,x^3+9\,x}\,\left (2\,x+3\right )+2880\,x^3+3200\,x^5-40\right )-40\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- 9*x - 24*x^3 - 16*x^5)*(exp(9*x + 24*x^3 + 16*x^5)*(x + 3) + log(x)*(360*x + exp(9*x + 24*x^3 + 16*
x^5)*(2*x + 3) + 2880*x^3 + 3200*x^5 - 40) - 40),x)

[Out]

int(-exp(- 9*x - 24*x^3 - 16*x^5)*(exp(9*x + 24*x^3 + 16*x^5)*(x + 3) + log(x)*(360*x + exp(9*x + 24*x^3 + 16*
x^5)*(2*x + 3) + 2880*x^3 + 3200*x^5 - 40) - 40), x)

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sympy [A]  time = 0.38, size = 34, normalized size = 1.26 \begin {gather*} 40 x e^{- 16 x^{5} - 24 x^{3} - 9 x} \log {\relax (x )} + \left (- x^{2} - 3 x\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-3)*exp(16*x**5+24*x**3+9*x)-3200*x**5-2880*x**3-360*x+40)*ln(x)+(-3-x)*exp(16*x**5+24*x**3+9
*x)+40)/exp(16*x**5+24*x**3+9*x),x)

[Out]

40*x*exp(-16*x**5 - 24*x**3 - 9*x)*log(x) + (-x**2 - 3*x)*log(x)

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