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

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

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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 = {} $$\begin{array}{c}\int \frac{e^{-1+12x}(4-48x)-x^2}{16e^{-2+24x}+8e^{-1+12x}{x}^2+x^4}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^2(e^{-1+12x}(4-48x)-x^2)}{{(4e^{12x}+e{x}^2)}^2}dx\\ & =e^2\int \frac{e^{-1+12x}(4-48x)-x^2}{{(4e^{12x}+e{x}^2)}^2}dx\\ & =e^2\int (\frac{2x^2(-1+6x)}{{(4e^{12x}+e{x}^2)}^2}-\frac{-1+12x}{e(4e^{12x}+e{x}^2)})dx\\ & =-(e\int \frac{-1+12x}{4e^{12x}+e{x}^2}dx)+\left(2e^2\right)\int \frac{x^2(-1+6x)}{{(4e^{12x}+e{x}^2)}^2}dx\\ & =-(e\int (-\frac{1}{4e^{12x}+e{x}^2}+\frac{12x}{4e^{12x}+e{x}^2})dx)+\left(2e^2\right)\int (-\frac{x^2}{{(4e^{12x}+e{x}^2)}^2}+\frac{6x^3}{{(4e^{12x}+e{x}^2)}^2})dx\\ & =e\int \frac{1}{4e^{12x}+e{x}^2}dx-(12e)\int \frac{x}{4e^{12x}+e{x}^2}dx-\left(2e^2\right)\int \frac{x^2}{{(4e^{12x}+e{x}^2)}^2}dx+\left(12e^2\right)\int \frac{x^3}{{(4e^{12x}+e{x}^2)}^2}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{x}{x^2+4e^{(12x-1)}}\end{array}$$

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 $$\begin{array}{c}\frac{xe}{x^{2}e+4e^{\left(12x\right)}}\end{array}$$

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	$\frac{x}{x^2+4{\mathrm{e}}^{12x-1}}$	$17$
risch	$\frac{x}{x^2+4{\mathrm{e}}^{12x-1}}$	$17$

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 $$\begin{array}{c}\frac{xe}{x^{2}e+4e^{\left(12x\right)}}\end{array}$$

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 $$\begin{array}{c}\frac{x}{4{\mathrm{e}}^{12x-1}+x^2}\end{array}$$

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 $$\begin{array}{c}\frac{x}{x^2+4e^{12x-1}}\end{array}$$

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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