3.33.63 \(\int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{\frac {4 e^{-1+12 x}}{x}+x} \]

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Rubi [F]  time = 1.00, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-1 + 12*x)*(4 - 48*x) - x^2)/(16*E^(-2 + 24*x) + 8*E^(-1 + 12*x)*x^2 + x^4),x]

[Out]

-2*E^2*Defer[Int][x^2/(4*E^(12*x) + E*x^2)^2, x] + 12*E^2*Defer[Int][x^3/(4*E^(12*x) + E*x^2)^2, x] + E*Defer[
Int][(4*E^(12*x) + E*x^2)^(-1), x] - 12*E*Defer[Int][x/(4*E^(12*x) + E*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 \left (e^{-1+12 x} (4-48 x)-x^2\right )}{\left (4 e^{12 x}+e x^2\right )^2} \, dx\\ &=e^2 \int \frac {e^{-1+12 x} (4-48 x)-x^2}{\left (4 e^{12 x}+e x^2\right )^2} \, dx\\ &=e^2 \int \left (\frac {2 x^2 (-1+6 x)}{\left (4 e^{12 x}+e x^2\right )^2}-\frac {-1+12 x}{e \left (4 e^{12 x}+e x^2\right )}\right ) \, dx\\ &=-\left (e \int \frac {-1+12 x}{4 e^{12 x}+e x^2} \, dx\right )+\left (2 e^2\right ) \int \frac {x^2 (-1+6 x)}{\left (4 e^{12 x}+e x^2\right )^2} \, dx\\ &=-\left (e \int \left (-\frac {1}{4 e^{12 x}+e x^2}+\frac {12 x}{4 e^{12 x}+e x^2}\right ) \, dx\right )+\left (2 e^2\right ) \int \left (-\frac {x^2}{\left (4 e^{12 x}+e x^2\right )^2}+\frac {6 x^3}{\left (4 e^{12 x}+e x^2\right )^2}\right ) \, dx\\ &=e \int \frac {1}{4 e^{12 x}+e x^2} \, dx-(12 e) \int \frac {x}{4 e^{12 x}+e x^2} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (4 e^{12 x}+e x^2\right )^2} \, dx+\left (12 e^2\right ) \int \frac {x^3}{\left (4 e^{12 x}+e x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 18, normalized size = 1.12 \begin {gather*} \frac {e x}{4 e^{12 x}+e x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 + 12*x)*(4 - 48*x) - x^2)/(16*E^(-2 + 24*x) + 8*E^(-1 + 12*x)*x^2 + x^4),x]

[Out]

(E*x)/(4*E^(12*x) + E*x^2)

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fricas [A]  time = 0.57, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 4 \, e^{\left (12 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="fricas")

[Out]

x/(x^2 + 4*e^(12*x - 1))

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giac [A]  time = 0.21, size = 19, normalized size = 1.19 \begin {gather*} \frac {x e}{x^{2} e + 4 \, e^{\left (12 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="giac")

[Out]

x*e/(x^2*e + 4*e^(12*x))

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maple [A]  time = 0.08, size = 17, normalized size = 1.06

method	result	size
norman	\(\frac {x}{x^{2}+4 \,{\mathrm e}^{12 x -1}}\)	\(17\)
risch	\(\frac {x}{x^{2}+4 \,{\mathrm e}^{12 x -1}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 19, normalized size = 1.19 \begin {gather*} \frac {x e}{x^{2} e + 4 \, e^{\left (12 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="maxima")

[Out]

x*e/(x^2*e + 4*e^(12*x))

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mupad [B]  time = 0.10, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{4\,{\mathrm {e}}^{12\,x-1}+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(12*x - 1)*(48*x - 4) + x^2)/(16*exp(24*x - 2) + 8*x^2*exp(12*x - 1) + x^4),x)

[Out]

x/(4*exp(12*x - 1) + x^2)

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sympy [A]  time = 0.13, size = 12, normalized size = 0.75 \begin {gather*} \frac {x}{x^{2} + 4 e^{12 x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-48*x+4)*exp(12*x-1)-x**2)/(16*exp(12*x-1)**2+8*x**2*exp(12*x-1)+x**4),x)

[Out]

x/(x**2 + 4*exp(12*x - 1))

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