3.85.31 $\int \frac{e^{2x}(-2+2e^5)+e^{2e^{5}x}(8e^{8x}+2x+e^{4x}(2+8x))}{-e^{2x}+e^{2e^{5}x}(-4+e^{8x}+2e^{4x}x+x^2)}dx$

Optimal. Leaf size=26 $$\log (-4-e^{2x-2e^{5}x}+{(e^{4x}+x)}^2)$$

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Rubi [F]  time = 13.70, 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 = {} $$\begin{array}{c}\int \frac{e^{2x}(-2+2e^5)+e^{2e^{5}x}(8e^{8x}+2x+e^{4x}(2+8x))}{-e^{2x}+e^{2e^{5}x}(-4+e^{8x}+2e^{4x}x+x^2)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*(2 + 8*x)))/(-E^(2*x) + E^(2*E^5*x)*(-4 + E
^(8*x) + 2*E^(4*x)*x + x^2)),x]

[Out]

Log[4 - E^(8*x) - 2*E^(4*x)*x - x^2] - 8*(1 - E^5)*Defer[Int][E^(2*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(
2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^(2*E^5*x)*x^2)), x] - 2*Defer[Int][E^(6*x
)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E
^(2*E^5*x)*x^2)), x] - 2*(3 + E^5)*Defer[Int][E^(10*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E
^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^(2*E^5*x)*x^2)), x] - 4*(1 + E^5)*Defer[Int][(E^(6*x)*x)
/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(E^(2*x) + 4*E^(2*E^5*x) - E^(2*(4 + E^5)*x) - 2*E^(2*(2 + E^5)*x)*x - E^
(2*E^5*x)*x^2)), x] + 2*Defer[Int][(E^(2*x)*x)/((-4 + E^(8*x) + 2*E^(4*x)*x + x^2)*(-E^(2*x) - 4*E^(2*E^5*x) +
E^(2*(4 + E^5)*x) + 2*E^(2*(2 + E^5)*x)*x + E^(2*E^5*x)*x^2)), x] - 2*(1 - E^5)*Defer[Int][(E^(2*x)*x^2)/((-4
+ E^(8*x) + 2*E^(4*x)*x + x^2)*(-E^(2*x) - 4*E^(2*E^5*x) + E^(2*(4 + E^5)*x) + 2*E^(2*(2 + E^5)*x)*x + E^(2*E
^5*x)*x^2)), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{2(1+4e^{4x})(e^{4x}+x)}{-4+e^{8x}+2e^{4x}x+x^2}+\frac{2e^{2x}(e^{4x}+4(1-e^5)+3e^{8x}(1+\frac{e^5}{3})+x+2e^{4x}(1+e^5)x-(1-e^5)x^2)}{(4-e^{8x}-2e^{4x}x-x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)})dx\\ & =2\int \frac{(1+4e^{4x})(e^{4x}+x)}{-4+e^{8x}+2e^{4x}x+x^2}dx+2\int \frac{e^{2x}(e^{4x}+4(1-e^5)+3e^{8x}(1+\frac{e^5}{3})+x+2e^{4x}(1+e^5)x-(1-e^5)x^2)}{(4-e^{8x}-2e^{4x}x-x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}dx\\ & =\log (4-e^{8x}-2e^{4x}x-x^2)+2\int (-\frac{e^{6x}}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}+\frac{4e^{2x}(-1+e^5)}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}-\frac{e^{10x}(3+e^5)}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}-\frac{2e^{6x}(1+e^5)x}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}+\frac{e^{2x}x}{(-4+e^{8x}+2e^{4x}x+x^2)(-e^{2x}-4e^{2e^{5}x}+e^{2(4+e^5)x}+2e^{2(2+e^5)x}x+e^{2e^{5}x}{x}^2)}+\frac{e^{2x}(-1+e^5)x^2}{(-4+e^{8x}+2e^{4x}x+x^2)(-e^{2x}-4e^{2e^{5}x}+e^{2(4+e^5)x}+2e^{2(2+e^5)x}x+e^{2e^{5}x}{x}^2)})dx\\ & =\log (4-e^{8x}-2e^{4x}x-x^2)-2\int \frac{e^{6x}}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}dx+2\int \frac{e^{2x}x}{(-4+e^{8x}+2e^{4x}x+x^2)(-e^{2x}-4e^{2e^{5}x}+e^{2(4+e^5)x}+2e^{2(2+e^5)x}x+e^{2e^{5}x}{x}^2)}dx-\left(2(1-e^5)\right)\int \frac{e^{2x}{x}^2}{(-4+e^{8x}+2e^{4x}x+x^2)(-e^{2x}-4e^{2e^{5}x}+e^{2(4+e^5)x}+2e^{2(2+e^5)x}x+e^{2e^{5}x}{x}^2)}dx-\left(8(1-e^5)\right)\int \frac{e^{2x}}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}dx-\left(4(1+e^5)\right)\int \frac{e^{6x}x}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}dx-\left(2(3+e^5)\right)\int \frac{e^{10x}}{(-4+e^{8x}+2e^{4x}x+x^2)(e^{2x}+4e^{2e^{5}x}-e^{2(4+e^5)x}-2e^{2(2+e^5)x}x-e^{2e^{5}x}{x}^2)}dx\end{array}\end{array}$$

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Mathematica [B]  time = 5.13, size = 65, normalized size = 2.50 $$\begin{array}{c}-2e^{5}x+\log (-e^{2x}-4e^{2e^{5}x}+e^{8x+2e^{5}x}+2e^{4x+2e^{5}x}x+e^{2e^{5}x}{x}^2)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*(2 + 8*x)))/(-E^(2*x) + E^(2*E^5*x)*(
-4 + E^(8*x) + 2*E^(4*x)*x + x^2)),x]

[Out]

-2*E^5*x + Log[-E^(2*x) - 4*E^(2*E^5*x) + E^(8*x + 2*E^5*x) + 2*E^(4*x + 2*E^5*x)*x + E^(2*E^5*x)*x^2]

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fricas [B]  time = 1.91, size = 73, normalized size = 2.81 $$\begin{array}{c}-2x{e}^5+\log (x^2+2x{e}^{\left(4x\right)}+e^{\left(8x\right)}-4)+\log \left(\frac{(x^2+2x{e}^{\left(4x\right)}+e^{\left(8x\right)}-4)e^{\left(2x{e}^5\right)}-e^{\left(2x\right)}}{x^2+2x{e}^{\left(4x\right)}+e^{\left(8x\right)}-4}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="fricas")

[Out]

-2*x*e^5 + log(x^2 + 2*x*e^(4*x) + e^(8*x) - 4) + log(((x^2 + 2*x*e^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) - e^(2*x)
)/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))

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giac [B]  time = 0.42, size = 55, normalized size = 2.12 $$\begin{array}{c}-2x{e}^5+\log (x^2{e}^{\left(2x{e}^5\right)}+2x{e}^{(2x{e}^5+4x)}-4e^{\left(2x{e}^5\right)}+e^{(2x{e}^5+8x)}-e^{\left(2x\right)})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="giac")

[Out]

-2*x*e^5 + log(x^2*e^(2*x*e^5) + 2*x*e^(2*x*e^5 + 4*x) - 4*e^(2*x*e^5) + e^(2*x*e^5 + 8*x) - e^(2*x))

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maple [B]  time = 0.07, size = 56, normalized size = 2.15

method	result	size
risch	$\ln ({\mathrm{e}}^{8x}+2x{\mathrm{e}}^{4x}+x^2-4)-2x{\mathrm{e}}^5+\ln ({\mathrm{e}}^{2x{\mathrm{e}}^5}-\frac{{\mathrm{e}}^{2x}}{{\mathrm{e}}^{8x}+2x{\mathrm{e}}^{4x}+x^2-4})$	$56$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-
4)*exp(x*exp(5))^2-exp(x)^2),x,method=_RETURNVERBOSE)

[Out]

ln(exp(8*x)+2*x*exp(4*x)+x^2-4)-2*x*exp(5)+ln(exp(2*x*exp(5))-exp(2*x)/(exp(8*x)+2*x*exp(4*x)+x^2-4))

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maxima [B]  time = 0.41, size = 72, normalized size = 2.77 $$\begin{array}{c}-2x{e}^5+\log (x+e^{\left(4x\right)}+2)+\log (x+e^{\left(4x\right)}-2)+\log \left(\frac{(x^2+2x{e}^{\left(4x\right)}+e^{\left(8x\right)}-4)e^{\left(2x{e}^5\right)}-e^{\left(2x\right)}}{x^2+2x{e}^{\left(4x\right)}+e^{\left(8x\right)}-4}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x
)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="maxima")

[Out]

-2*x*e^5 + log(x + e^(4*x) + 2) + log(x + e^(4*x) - 2) + log(((x^2 + 2*x*e^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) -
e^(2*x))/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))

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mupad [B]  time = 0.58, size = 92, normalized size = 3.54 $$\begin{array}{c}\ln \left(\frac{{\mathrm{e}}^{8x}{\mathrm{e}}^{2x{\mathrm{e}}^5}-4{\mathrm{e}}^{2x{\mathrm{e}}^5}-{\mathrm{e}}^{2x}+x^2{\mathrm{e}}^{2x{\mathrm{e}}^5}+2x{\mathrm{e}}^{4x}{\mathrm{e}}^{2x{\mathrm{e}}^5}}{{\mathrm{e}}^{8x}+2x{\mathrm{e}}^{4x}+x^2-4}\right)+\ln ({\mathrm{e}}^{8x}+2x{\mathrm{e}}^{4x}+x^2-4)-2x{\mathrm{e}}^5\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x*exp(5))*(2*x + 8*exp(8*x) + exp(4*x)*(8*x + 2)) + exp(2*x)*(2*exp(5) - 2))/(exp(2*x) - exp(2*x*e
xp(5))*(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)),x)

[Out]

log((exp(8*x)*exp(2*x*exp(5)) - 4*exp(2*x*exp(5)) - exp(2*x) + x^2*exp(2*x*exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(
5)))/(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)) + log(exp(8*x) + 2*x*exp(4*x) + x^2 - 4) - 2*x*exp(5)

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sympy [B]  time = 2.89, size = 65, normalized size = 2.50 $$\begin{array}{c}-2x{e}^5+\log (x^2{e}^{2x{e}^5}+2x{e}^{4x}{e}^{2x{e}^5}+e^{8x}{e}^{2x{e}^5}-e^{2x}-4e^{2x{e}^5})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(4*x)**2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))**2+(2*exp(5)-2)*exp(x)**2)/((exp(4*x)**2+2*x*exp
(4*x)+x**2-4)*exp(x*exp(5))**2-exp(x)**2),x)

[Out]

-2*x*exp(5) + log(x**2*exp(2*x*exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(5)) + exp(8*x)*exp(2*x*exp(5)) - exp(2*x) -
4*exp(2*x*exp(5)))

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