3.90.93 \(\int \frac {e^{\frac {1}{4} (-12-x^3-4 x^4+4 \log (\frac {e^{-x} x}{10}))} (4-4 x-3 x^3-16 x^4)}{4 x} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{10} e^{-3-x-x^3 \left (\frac {1}{4}+x\right )} x \]

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Rubi [B]  time = 0.14, antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 4, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 2274, 2288} \begin {gather*} \frac {e^{\frac {1}{4} \left (-4 x^4-x^3-12\right )-x} \left (16 x^4+3 x^3+4 x\right )}{10 \left (16 x^3+3 x^2+4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-12 - x^3 - 4*x^4 + 4*Log[x/(10*E^x)])/4)*(4 - 4*x - 3*x^3 - 16*x^4))/(4*x),x]

[Out]

(E^(-x + (-12 - x^3 - 4*x^4)/4)*(4*x + 3*x^3 + 16*x^4))/(10*(4 + 3*x^2 + 16*x^3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

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 {1}{4} \int \frac {e^{\frac {1}{4} \left (-12-x^3-4 x^4+4 \log \left (\frac {e^{-x} x}{10}\right )\right )} \left (4-4 x-3 x^3-16 x^4\right )}{x} \, dx\\ &=\frac {1}{4} \int \frac {1}{10} e^{-x+\frac {1}{4} \left (-12-x^3-4 x^4\right )} \left (4-4 x-3 x^3-16 x^4\right ) \, dx\\ &=\frac {1}{40} \int e^{-x+\frac {1}{4} \left (-12-x^3-4 x^4\right )} \left (4-4 x-3 x^3-16 x^4\right ) \, dx\\ &=\frac {e^{-x+\frac {1}{4} \left (-12-x^3-4 x^4\right )} \left (4 x+3 x^3+16 x^4\right )}{10 \left (4+3 x^2+16 x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 24, normalized size = 1.09 \begin {gather*} \frac {1}{10} e^{-3-x-\frac {x^3}{4}-x^4} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-12 - x^3 - 4*x^4 + 4*Log[x/(10*E^x)])/4)*(4 - 4*x - 3*x^3 - 16*x^4))/(4*x),x]

[Out]

(E^(-3 - x - x^3/4 - x^4)*x)/10

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fricas [A]  time = 0.76, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (-x^{4} - \frac {1}{4} \, x^{3} + \log \left (\frac {1}{10} \, x e^{\left (-x\right )}\right ) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-16*x^4-3*x^3-4*x+4)*exp(log(1/10*x/exp(x))-x^4-1/4*x^3-3)/x,x, algorithm="fricas")

[Out]

e^(-x^4 - 1/4*x^3 + log(1/10*x*e^(-x)) - 3)

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giac [A]  time = 0.15, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (-x^{4} - \frac {1}{4} \, x^{3} + \log \left (\frac {1}{10} \, x e^{\left (-x\right )}\right ) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-16*x^4-3*x^3-4*x+4)*exp(log(1/10*x/exp(x))-x^4-1/4*x^3-3)/x,x, algorithm="giac")

[Out]

e^(-x^4 - 1/4*x^3 + log(1/10*x*e^(-x)) - 3)

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maple [A]  time = 0.09, size = 22, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{\ln \left (\frac {x \,{\mathrm e}^{-x}}{10}\right )-x^{4}-\frac {x^{3}}{4}-3}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-16*x^4-3*x^3-4*x+4)*exp(ln(1/10*x/exp(x))-x^4-1/4*x^3-3)/x,x,method=_RETURNVERBOSE)

[Out]

exp(ln(1/10*x/exp(x))-x^4-1/4*x^3-3)

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maxima [A]  time = 0.41, size = 19, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, x e^{\left (-x^{4} - \frac {1}{4} \, x^{3} - x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-16*x^4-3*x^3-4*x+4)*exp(log(1/10*x/exp(x))-x^4-1/4*x^3-3)/x,x, algorithm="maxima")

[Out]

1/10*x*e^(-x^4 - 1/4*x^3 - x - 3)

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mupad [B]  time = 5.47, size = 21, normalized size = 0.95 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{-\frac {x^3}{4}}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log((x*exp(-x))/10) - x^3/4 - x^4 - 3)*(4*x + 3*x^3 + 16*x^4 - 4))/(4*x),x)

[Out]

(x*exp(-x)*exp(-3)*exp(-x^4)*exp(-x^3/4))/10

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sympy [A]  time = 0.27, size = 19, normalized size = 0.86 \begin {gather*} \frac {x e^{- x} e^{- x^{4} - \frac {x^{3}}{4} - 3}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-16*x**4-3*x**3-4*x+4)*exp(ln(1/10*x/exp(x))-x**4-1/4*x**3-3)/x,x)

[Out]

x*exp(-x)*exp(-x**4 - x**3/4 - 3)/10

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