3.41.42 $\int \frac{e^{\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}}(-48-96x-12x^2+e^{\frac{1}{3}(-e^x+3x)}(-36-12x+e^x(16+4x)))}{16+8x+x^2}dx$

Optimal. Leaf size=31 $$e^{-4+\frac{12(4-e^{-\frac{e^x}{3}+x}-x^2)}{4+x}}$$

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Rubi [F]  time = 6.99, 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 = {} $$\begin{array}{c}\int \frac{\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)(-48-96x-12x^2+e^{\frac{1}{3}(-e^x+3x)}(-36-12x+e^x(16+4x)))}{16+8x+x^2}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^((32 - 12*E^((-E^x + 3*x)/3) - 4*x - 12*x^2)/(4 + x))*(-48 - 96*x - 12*x^2 + E^((-E^x + 3*x)/3)*(-36 -
12*x + E^x*(16 + 4*x))))/(16 + 8*x + x^2),x]

[Out]

-12*Defer[Int][E^((32 - 12*E^((-E^x + 3*x)/3) - 4*x - 12*x^2)/(4 + x)), x] + 144*Defer[Int][E^((32 - 12*E^((-E
^x + 3*x)/3) - 4*x - 12*x^2)/(4 + x))/(4 + x)^2, x] + 12*Defer[Int][E^(-1/3*E^x + x + (32 - 12*E^((-E^x + 3*x)
/3) - 4*x - 12*x^2)/(4 + x))/(4 + x)^2, x] - 12*Defer[Int][E^(-1/3*E^x + x + (32 - 12*E^((-E^x + 3*x)/3) - 4*x
- 12*x^2)/(4 + x))/(4 + x), x] + 4*Defer[Int][E^((-E^x + 6*x)/3 + (32 - 12*E^((-E^x + 3*x)/3) - 4*x - 12*x^2)
/(4 + x))/(4 + x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)(-48-96x-12x^2+e^{\frac{1}{3}(-e^x+3x)}(-36-12x+e^x(16+4x)))}{(4+x{)}^2}dx\\ & =\int (-\frac{12\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})(3+x)}{(4+x{)}^2}+\frac{4\exp (\frac{1}{3}(-e^x+6x)+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x}-\frac{12\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)(4+8x+x^2)}{(4+x{)}^2})dx\\ & =4\int \frac{\exp (\frac{1}{3}(-e^x+6x)+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x}dx-12\int \frac{\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})(3+x)}{(4+x{)}^2}dx-12\int \frac{\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)(4+8x+x^2)}{(4+x{)}^2}dx\\ & =4\int \frac{\exp (\frac{1}{3}(-e^x+6x)+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x}dx-12\int (\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)-\frac{12\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)}{(4+x{)}^2})dx-12\int (-\frac{\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{(4+x{)}^2}+\frac{\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x})dx\\ & =4\int \frac{\exp (\frac{1}{3}(-e^x+6x)+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x}dx-12\int \exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)dx+12\int \frac{\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{(4+x{)}^2}dx-12\int \frac{\exp (-\frac{e^x}{3}+x+\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x})}{4+x}dx+144\int \frac{\exp \left(\frac{32-12e^{\frac{1}{3}(-e^x+3x)}-4x-12x^2}{4+x}\right)}{(4+x{)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.21, size = 32, normalized size = 1.03 $$\begin{array}{c}{e}^{44-12x-\frac{144}{4+x}-\frac{12e^{-\frac{e^x}{3}+x}}{4+x}}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^((32 - 12*E^((-E^x + 3*x)/3) - 4*x - 12*x^2)/(4 + x))*(-48 - 96*x - 12*x^2 + E^((-E^x + 3*x)/3)*(
-36 - 12*x + E^x*(16 + 4*x))))/(16 + 8*x + x^2),x]

[Out]

E^(44 - 12*x - 144/(4 + x) - (12*E^(-1/3*E^x + x))/(4 + x))

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fricas [A]  time = 0.82, size = 25, normalized size = 0.81 $$\begin{array}{c}{e}^{(-\frac{4(3x^2+x+3e^{(x-\frac{1}{3}{e}^x)}-8)}{x+4})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+16)*exp(x)-12*x-36)*exp(-1/3*exp(x)+x)-12*x^2-96*x-48)*exp((-12*exp(-1/3*exp(x)+x)-12*x^2-4*x
+32)/(4+x))/(x^2+8*x+16),x, algorithm="fricas")

[Out]

e^(-4*(3*x^2 + x + 3*e^(x - 1/3*e^x) - 8)/(x + 4))

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giac [A]  time = 0.44, size = 41, normalized size = 1.32 $$\begin{array}{c}{e}^{(-\frac{12x^2}{x+4}-\frac{4x}{x+4}-\frac{12e^{(x-\frac{1}{3}{e}^x)}}{x+4}+\frac{32}{x+4})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+16)*exp(x)-12*x-36)*exp(-1/3*exp(x)+x)-12*x^2-96*x-48)*exp((-12*exp(-1/3*exp(x)+x)-12*x^2-4*x
+32)/(4+x))/(x^2+8*x+16),x, algorithm="giac")

[Out]

e^(-12*x^2/(x + 4) - 4*x/(x + 4) - 12*e^(x - 1/3*e^x)/(x + 4) + 32/(x + 4))

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maple [A]  time = 0.16, size = 26, normalized size = 0.84

method	result	size
risch	${\mathrm{e}}^{-\frac{4(3x^2+3{\mathrm{e}}^{-\frac{{\mathrm{e}}^x}{3}+x}+x-8)}{4+x}}$	$26$
norman	$\frac{x{\mathrm{e}}^{\frac{-12{\mathrm{e}}^{-\frac{{\mathrm{e}}^x}{3}+x}-12x^2-4x+32}{4+x}}+4{\mathrm{e}}^{\frac{-12{\mathrm{e}}^{-\frac{{\mathrm{e}}^x}{3}+x}-12x^2-4x+32}{4+x}}}{4+x}$	$64$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x+16)*exp(x)-12*x-36)*exp(-1/3*exp(x)+x)-12*x^2-96*x-48)*exp((-12*exp(-1/3*exp(x)+x)-12*x^2-4*x+32)/(
4+x))/(x^2+8*x+16),x,method=_RETURNVERBOSE)

[Out]

exp(-4*(3*x^2+3*exp(-1/3*exp(x)+x)+x-8)/(4+x))

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maxima [A]  time = 0.59, size = 27, normalized size = 0.87 $$\begin{array}{c}{e}^{(-12x-\frac{12e^{(x-\frac{1}{3}{e}^x)}}{x+4}-\frac{144}{x+4}+44)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+16)*exp(x)-12*x-36)*exp(-1/3*exp(x)+x)-12*x^2-96*x-48)*exp((-12*exp(-1/3*exp(x)+x)-12*x^2-4*x
+32)/(4+x))/(x^2+8*x+16),x, algorithm="maxima")

[Out]

e^(-12*x - 12*e^(x - 1/3*e^x)/(x + 4) - 144/(x + 4) + 44)

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mupad [B]  time = 3.19, size = 44, normalized size = 1.42 $$\begin{array}{c}{\mathrm{e}}^{-\frac{4x}{x+4}}{\mathrm{e}}^{-\frac{12{\mathrm{e}}^{-\frac{{\mathrm{e}}^x}{3}}{\mathrm{e}}^x}{x+4}}{\mathrm{e}}^{-\frac{12x^2}{x+4}}{\mathrm{e}}^{\frac{32}{x+4}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(4*x + 12*exp(x - exp(x)/3) + 12*x^2 - 32)/(x + 4))*(96*x + exp(x - exp(x)/3)*(12*x - exp(x)*(4*x +
16) + 36) + 12*x^2 + 48))/(8*x + x^2 + 16),x)

[Out]

exp(-(4*x)/(x + 4))*exp(-(12*exp(-exp(x)/3)*exp(x))/(x + 4))*exp(-(12*x^2)/(x + 4))*exp(32/(x + 4))

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sympy [A]  time = 0.78, size = 24, normalized size = 0.77 $$\begin{array}{c}{e}^{\frac{-12x^2-4x-12e^{x-\frac{e^x}{3}}+32}{x+4}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+16)*exp(x)-12*x-36)*exp(-1/3*exp(x)+x)-12*x**2-96*x-48)*exp((-12*exp(-1/3*exp(x)+x)-12*x**2-4
*x+32)/(4+x))/(x**2+8*x+16),x)

[Out]

exp((-12*x**2 - 4*x - 12*exp(x - exp(x)/3) + 32)/(x + 4))

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