3.24.71 \(\int \frac {e^{-4-\frac {3-72 x+414 x^2+204 x^3+171 x^4+36 x^5+12 x^6}{e^4 (1-4 x+4 x^2)}} (-60 x+684 x^2+612 x^3+276 x^4-504 x^5-144 x^6-96 x^7+e^4 (1-6 x+12 x^2-8 x^3))}{-x^2+6 x^3-12 x^4+8 x^5} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^{-\frac {3 \left (\frac {24}{4-\frac {2}{x}}+(1+x)^2\right )^2}{e^4}}}{x} \]

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Rubi [B]  time = 2.22, antiderivative size = 204, normalized size of antiderivative = 6.80, number of steps used = 1, number of rules used = 1, integrand size = 128, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2288} \begin {gather*} \frac {\left (8 x^7+12 x^6+42 x^5-23 x^4-51 x^3-57 x^2+5 x\right ) \exp \left (-\frac {3 \left (4 x^6+12 x^5+57 x^4+68 x^3+138 x^2-24 x+1\right )}{e^4 \left (4 x^2-4 x+1\right )}-4\right )}{\left (-8 x^5+12 x^4-6 x^3+x^2\right ) \left (\frac {3 \left (-2 x^5-5 x^4-19 x^3-17 x^2-23 x+2\right )}{e^4 \left (4 x^2-4 x+1\right )}-\frac {(1-2 x) \left (4 x^6+12 x^5+57 x^4+68 x^3+138 x^2-24 x+1\right )}{e^4 \left (4 x^2-4 x+1\right )^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-4 - (3 - 72*x + 414*x^2 + 204*x^3 + 171*x^4 + 36*x^5 + 12*x^6)/(E^4*(1 - 4*x + 4*x^2)))*(-60*x + 684*
x^2 + 612*x^3 + 276*x^4 - 504*x^5 - 144*x^6 - 96*x^7 + E^4*(1 - 6*x + 12*x^2 - 8*x^3)))/(-x^2 + 6*x^3 - 12*x^4
+ 8*x^5),x]

[Out]

(E^(-4 - (3*(1 - 24*x + 138*x^2 + 68*x^3 + 57*x^4 + 12*x^5 + 4*x^6))/(E^4*(1 - 4*x + 4*x^2)))*(5*x - 57*x^2 -
51*x^3 - 23*x^4 + 42*x^5 + 12*x^6 + 8*x^7))/((x^2 - 6*x^3 + 12*x^4 - 8*x^5)*((3*(2 - 23*x - 17*x^2 - 19*x^3 -
5*x^4 - 2*x^5))/(E^4*(1 - 4*x + 4*x^2)) - ((1 - 2*x)*(1 - 24*x + 138*x^2 + 68*x^3 + 57*x^4 + 12*x^5 + 4*x^6))/
(E^4*(1 - 4*x + 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\exp \left (-4-\frac {3 \left (1-24 x+138 x^2+68 x^3+57 x^4+12 x^5+4 x^6\right )}{e^4 \left (1-4 x+4 x^2\right )}\right ) \left (5 x-57 x^2-51 x^3-23 x^4+42 x^5+12 x^6+8 x^7\right )}{\left (x^2-6 x^3+12 x^4-8 x^5\right ) \left (\frac {3 \left (2-23 x-17 x^2-19 x^3-5 x^4-2 x^5\right )}{e^4 \left (1-4 x+4 x^2\right )}-\frac {(1-2 x) \left (1-24 x+138 x^2+68 x^3+57 x^4+12 x^5+4 x^6\right )}{e^4 \left (1-4 x+4 x^2\right )^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 35, normalized size = 1.17 \begin {gather*} \frac {e^{-\frac {3 \left (-1+12 x+3 x^2+2 x^3\right )^2}{e^4 (1-2 x)^2}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 - (3 - 72*x + 414*x^2 + 204*x^3 + 171*x^4 + 36*x^5 + 12*x^6)/(E^4*(1 - 4*x + 4*x^2)))*(-60*x
+ 684*x^2 + 612*x^3 + 276*x^4 - 504*x^5 - 144*x^6 - 96*x^7 + E^4*(1 - 6*x + 12*x^2 - 8*x^3)))/(-x^2 + 6*x^3 -
12*x^4 + 8*x^5),x]

[Out]

1/(E^((3*(-1 + 12*x + 3*x^2 + 2*x^3)^2)/(E^4*(1 - 2*x)^2))*x)

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fricas [B]  time = 0.53, size = 67, normalized size = 2.23 \begin {gather*} \frac {e^{\left (-\frac {{\left (12 \, x^{6} + 36 \, x^{5} + 171 \, x^{4} + 204 \, x^{3} + 414 \, x^{2} + 4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} e^{4} - 72 \, x + 3\right )} e^{\left (-4\right )}}{4 \, x^{2} - 4 \, x + 1} + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3+12*x^2-6*x+1)*exp(4)-96*x^7-144*x^6-504*x^5+276*x^4+612*x^3+684*x^2-60*x)/(8*x^5-12*x^4+6*x
^3-x^2)/exp(4)/exp((12*x^6+36*x^5+171*x^4+204*x^3+414*x^2-72*x+3)/(4*x^2-4*x+1)/exp(4)),x, algorithm="fricas")

[Out]

e^(-(12*x^6 + 36*x^5 + 171*x^4 + 204*x^3 + 414*x^2 + 4*(4*x^2 - 4*x + 1)*e^4 - 72*x + 3)*e^(-4)/(4*x^2 - 4*x +
1) + 4)/x

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giac [B]  time = 1.35, size = 58, normalized size = 1.93 \begin {gather*} \frac {e^{\left (-\frac {3 \, {\left (4 \, x^{6} + 12 \, x^{5} + 57 \, x^{4} + 68 \, x^{3} + 134 \, x^{2} - 20 \, x\right )}}{4 \, x^{2} e^{4} - 4 \, x e^{4} + e^{4}} - 3 \, e^{\left (-4\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3+12*x^2-6*x+1)*exp(4)-96*x^7-144*x^6-504*x^5+276*x^4+612*x^3+684*x^2-60*x)/(8*x^5-12*x^4+6*x
^3-x^2)/exp(4)/exp((12*x^6+36*x^5+171*x^4+204*x^3+414*x^2-72*x+3)/(4*x^2-4*x+1)/exp(4)),x, algorithm="giac")

[Out]

e^(-3*(4*x^6 + 12*x^5 + 57*x^4 + 68*x^3 + 134*x^2 - 20*x)/(4*x^2*e^4 - 4*x*e^4 + e^4) - 3*e^(-4))/x

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maple [A]  time = 0.40, size = 34, normalized size = 1.13

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {3 \left (2 x^{3}+3 x^{2}+12 x -1\right )^{2} {\mathrm e}^{-4}}{\left (2 x -1\right )^{2}}}}{x}\)	\(34\)
gosper	\(\frac {{\mathrm e}^{-\frac {3 \left (4 x^{6}+12 x^{5}+57 x^{4}+68 x^{3}+138 x^{2}-24 x +1\right ) {\mathrm e}^{-4}}{4 x^{2}-4 x +1}}}{x}\)	\(56\)
norman	\(\frac {\left (4 x^{2}-4 x +1\right ) {\mathrm e}^{-\frac {\left (12 x^{6}+36 x^{5}+171 x^{4}+204 x^{3}+414 x^{2}-72 x +3\right ) {\mathrm e}^{-4}}{4 x^{2}-4 x +1}}}{x \left (2 x -1\right )^{2}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^3+12*x^2-6*x+1)*exp(4)-96*x^7-144*x^6-504*x^5+276*x^4+612*x^3+684*x^2-60*x)/(8*x^5-12*x^4+6*x^3-x^2
)/exp(4)/exp((12*x^6+36*x^5+171*x^4+204*x^3+414*x^2-72*x+3)/(4*x^2-4*x+1)/exp(4)),x,method=_RETURNVERBOSE)

[Out]

1/x*exp(-3*(2*x^3+3*x^2+12*x-1)^2*exp(-4)/(2*x-1)^2)

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maxima [B]  time = 1.58, size = 69, normalized size = 2.30 \begin {gather*} \frac {e^{\left (-3 \, x^{4} e^{\left (-4\right )} - 12 \, x^{3} e^{\left (-4\right )} - 54 \, x^{2} e^{\left (-4\right )} - 102 \, x e^{\left (-4\right )} - \frac {108}{4 \, x^{2} e^{4} - 4 \, x e^{4} + e^{4}} - \frac {297}{2 \, x e^{4} - e^{4}} - 192 \, e^{\left (-4\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3+12*x^2-6*x+1)*exp(4)-96*x^7-144*x^6-504*x^5+276*x^4+612*x^3+684*x^2-60*x)/(8*x^5-12*x^4+6*x
^3-x^2)/exp(4)/exp((12*x^6+36*x^5+171*x^4+204*x^3+414*x^2-72*x+3)/(4*x^2-4*x+1)/exp(4)),x, algorithm="maxima")

[Out]

e^(-3*x^4*e^(-4) - 12*x^3*e^(-4) - 54*x^2*e^(-4) - 102*x*e^(-4) - 108/(4*x^2*e^4 - 4*x*e^4 + e^4) - 297/(2*x*e
^4 - e^4) - 192*e^(-4))/x

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mupad [B]  time = 1.82, size = 139, normalized size = 4.63 \begin {gather*} \frac {{\mathrm {e}}^{\frac {72\,x\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {12\,x^6\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {36\,x^5\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {171\,x^4\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {204\,x^3\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {414\,x^2\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}\,{\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^{-4}}{4\,x^2-4\,x+1}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp(-4)*(414*x^2 - 72*x + 204*x^3 + 171*x^4 + 36*x^5 + 12*x^6 + 3))/(4*x^2 - 4*x + 1))*exp(-4)*(60*
x + exp(4)*(6*x - 12*x^2 + 8*x^3 - 1) - 684*x^2 - 612*x^3 - 276*x^4 + 504*x^5 + 144*x^6 + 96*x^7))/(x^2 - 6*x^
3 + 12*x^4 - 8*x^5),x)

[Out]

(exp((72*x*exp(-4))/(4*x^2 - 4*x + 1))*exp(-(12*x^6*exp(-4))/(4*x^2 - 4*x + 1))*exp(-(36*x^5*exp(-4))/(4*x^2 -
4*x + 1))*exp(-(171*x^4*exp(-4))/(4*x^2 - 4*x + 1))*exp(-(204*x^3*exp(-4))/(4*x^2 - 4*x + 1))*exp(-(414*x^2*e
xp(-4))/(4*x^2 - 4*x + 1))*exp(-(3*exp(-4))/(4*x^2 - 4*x + 1)))/x

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sympy [B]  time = 0.50, size = 48, normalized size = 1.60 \begin {gather*} \frac {e^{- \frac {12 x^{6} + 36 x^{5} + 171 x^{4} + 204 x^{3} + 414 x^{2} - 72 x + 3}{\left (4 x^{2} - 4 x + 1\right ) e^{4}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**3+12*x**2-6*x+1)*exp(4)-96*x**7-144*x**6-504*x**5+276*x**4+612*x**3+684*x**2-60*x)/(8*x**5-1
2*x**4+6*x**3-x**2)/exp(4)/exp((12*x**6+36*x**5+171*x**4+204*x**3+414*x**2-72*x+3)/(4*x**2-4*x+1)/exp(4)),x)

[Out]

exp(-(12*x**6 + 36*x**5 + 171*x**4 + 204*x**3 + 414*x**2 - 72*x + 3)*exp(-4)/(4*x**2 - 4*x + 1))/x

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