3.73.34 $\int \frac{27+3e^{4x}+18x+3x^2+e(-3x-82x^2)+e^{2x}(18+6x+e(-6x-164x^2))}{27x+756x^2+495x^3+82x^4+e^{4x}(3x+82x^2)+e^{2x}(18x+498x^2+164x^3)}dx$

Optimal. Leaf size=24 $$\frac{e}{3+e^{2x}+x}+\log \left(\frac{3x}{3+82x}\right)$$

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Rubi [F]  time = 1.21, 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 = {} $$\begin{array}{c}\int \frac{27+3e^{4x}+18x+3x^2+e(-3x-82x^2)+e^{2x}(18+6x+e(-6x-164x^2))}{27x+756x^2+495x^3+82x^4+e^{4x}(3x+82x^2)+e^{2x}(18x+498x^2+164x^3)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x + 756*
x^2 + 495*x^3 + 82*x^4 + E^(4*x)*(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]

[Out]

Log[x] - Log[3 + 82*x] + 5*E*Defer[Int][(3 + E^(2*x) + x)^(-2), x] + 2*E*Defer[Int][x/(3 + E^(2*x) + x)^2, x]
- 2*E*Defer[Int][(3 + E^(2*x) + x)^(-1), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{3e^{4x}+6e^{2x}(3+x)+3(3+x{)}^2-ex(3+82x)-2e^{1+2x}x(3+82x)}{x{(3+e^{2x}+x)}^2(3+82x)}dx\\ & =\int (-\frac{2e}{3+e^{2x}+x}+\frac{e(5+2x)}{{(3+e^{2x}+x)}^2}+\frac{3}{x(3+82x)})dx\\ & =3\int \frac{1}{x(3+82x)}dx+e\int \frac{5+2x}{{(3+e^{2x}+x)}^2}dx-(2e)\int \frac{1}{3+e^{2x}+x}dx\\ & =-(82\int \frac{1}{3+82x}dx)+e\int (\frac{5}{{(3+e^{2x}+x)}^2}+\frac{2x}{{(3+e^{2x}+x)}^2})dx-(2e)\int \frac{1}{3+e^{2x}+x}dx+\int \frac{1}{x}dx\\ & =\log (x)-\log (3+82x)+(2e)\int \frac{x}{{(3+e^{2x}+x)}^2}dx-(2e)\int \frac{1}{3+e^{2x}+x}dx+(5e)\int \frac{1}{{(3+e^{2x}+x)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.21, size = 23, normalized size = 0.96 $$\begin{array}{c}\frac{e}{3+e^{2x}+x}+\log (x)-\log (3+82x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x
+ 756*x^2 + 495*x^3 + 82*x^4 + E^(4*x)*(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]

[Out]

E/(3 + E^(2*x) + x) + Log[x] - Log[3 + 82*x]

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fricas [A]  time = 0.70, size = 41, normalized size = 1.71 $$\begin{array}{c}-\frac{(x+e^{\left(2x\right)}+3)\log (82x+3)-(x+e^{\left(2x\right)}+3)\log (x)-e}{x+e^{\left(2x\right)}+3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="fricas")

[Out]

-((x + e^(2*x) + 3)*log(82*x + 3) - (x + e^(2*x) + 3)*log(x) - e)/(x + e^(2*x) + 3)

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giac [B]  time = 0.23, size = 60, normalized size = 2.50 $$\begin{array}{c}-\frac{x\log (82x+3)+e^{\left(2x\right)}\log (82x+3)-x\log (x)-e^{\left(2x\right)}\log (x)-e+3\log (82x+3)-3\log (x)}{x+e^{\left(2x\right)}+3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="giac")

[Out]

-(x*log(82*x + 3) + e^(2*x)*log(82*x + 3) - x*log(x) - e^(2*x)*log(x) - e + 3*log(82*x + 3) - 3*log(x))/(x + e
^(2*x) + 3)

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maple [A]  time = 0.15, size = 24, normalized size = 1.00

method	result	size
norman	$\frac{\mathrm{e}}{{\mathrm{e}}^{2x}+3+x}-\ln (82x+3)+\ln (x)$	$24$
risch	$\frac{\mathrm{e}}{{\mathrm{e}}^{2x}+3+x}-\ln (82x+3)+\ln (x)$	$24$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(
x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x,method=_RETURNVERBOSE)

[Out]

exp(1)/(exp(x)^2+3+x)-ln(82*x+3)+ln(x)

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maxima [A]  time = 0.41, size = 23, normalized size = 0.96 $$\begin{array}{c}\frac{e}{x+e^{\left(2x\right)}+3}-\log (82x+3)+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="maxima")

[Out]

e/(x + e^(2*x) + 3) - log(82*x + 3) + log(x)

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mupad [B]  time = 4.43, size = 34, normalized size = 1.42 $$\begin{array}{c}\ln (x)-\ln (x+\frac{3}{82})-\frac{\frac{{\mathrm{e}}^{2x+1}}{3}+\frac{x\mathrm{e}}{3}}{x+{\mathrm{e}}^{2x}+3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + 3*exp(4*x) + exp(2*x)*(6*x - exp(1)*(6*x + 164*x^2) + 18) - exp(1)*(3*x + 82*x^2) + 3*x^2 + 27)/(2
7*x + exp(4*x)*(3*x + 82*x^2) + exp(2*x)*(18*x + 498*x^2 + 164*x^3) + 756*x^2 + 495*x^3 + 82*x^4),x)

[Out]

log(x) - log(x + 3/82) - (exp(2*x + 1)/3 + (x*exp(1))/3)/(x + exp(2*x) + 3)

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sympy [A]  time = 0.21, size = 20, normalized size = 0.83 $$\begin{array}{c}\log (x)-\log (x+\frac{3}{82})+\frac{e}{x+e^{2x}+3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)**4+((-164*x**2-6*x)*exp(1)+18+6*x)*exp(x)**2+(-82*x**2-3*x)*exp(1)+3*x**2+18*x+27)/((82*x*
*2+3*x)*exp(x)**4+(164*x**3+498*x**2+18*x)*exp(x)**2+82*x**4+495*x**3+756*x**2+27*x),x)

[Out]

log(x) - log(x + 3/82) + E/(x + exp(2*x) + 3)

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