\(\int \frac {27+3 e^{4 x}+18 x+3 x^2+e (-3 x-82 x^2)+e^{2 x} (18+6 x+e (-6 x-164 x^2))}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} (3 x+82 x^2)+e^{2 x} (18 x+498 x^2+164 x^3)} \, dx\) [7234]

Optimal result

Integrand size = 107, antiderivative size = 24 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{3+e^{2 x}+x}+\log \left (\frac {3 x}{3+82 x}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx \]

[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{align*} \text {integral}& = \int \frac {3 e^{4 x}+6 e^{2 x} (3+x)+3 (3+x)^2-e x (3+82 x)-2 e^{1+2 x} x (3+82 x)}{x \left (3+e^{2 x}+x\right )^2 (3+82 x)} \, dx \\ & = \int \left (-\frac {2 e}{3+e^{2 x}+x}+\frac {e (5+2 x)}{\left (3+e^{2 x}+x\right )^2}+\frac {3}{x (3+82 x)}\right ) \, dx \\ & = 3 \int \frac {1}{x (3+82 x)} \, dx+e \int \frac {5+2 x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx \\ & = -\left (82 \int \frac {1}{3+82 x} \, dx\right )+e \int \left (\frac {5}{\left (3+e^{2 x}+x\right )^2}+\frac {2 x}{\left (3+e^{2 x}+x\right )^2}\right ) \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+\int \frac {1}{x} \, dx \\ & = \log (x)-\log (3+82 x)+(2 e) \int \frac {x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+(5 e) \int \frac {1}{\left (3+e^{2 x}+x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{3+e^{2 x}+x}+\log (x)-\log (3+82 x) \]

[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]

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \left (x \right )\)	\(24\)
risch	\(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \left (x \right )\)	\(24\)
parallelrisch	\(\frac {{\mathrm e}^{2 x} \ln \left (x \right )-\ln \left (x +\frac {3}{82}\right ) {\mathrm e}^{2 x}+x \ln \left (x \right )-\ln \left (x +\frac {3}{82}\right ) x +{\mathrm e}+3 \ln \left (x \right )-3 \ln \left (x +\frac {3}{82}\right )}{{\mathrm e}^{2 x}+3+x}\)	\(52\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=-\frac {{\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (82 \, x + 3\right ) - {\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (x\right ) - e}{x + e^{\left (2 \, x\right )} + 3} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\log {\left (x \right )} - \log {\left (x + \frac {3}{82} \right )} + \frac {e}{x + e^{2 x} + 3} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{x + e^{\left (2 \, x\right )} + 3} - \log \left (82 \, x + 3\right ) + \log \left (x\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {x \log \left (2 \, x\right ) + e^{\left (2 \, x\right )} \log \left (2 \, x\right ) - x \log \left (82 \, x + 3\right ) - e^{\left (2 \, x\right )} \log \left (82 \, x + 3\right ) + e + 3 \, \log \left (2 \, x\right ) - 3 \, \log \left (82 \, x + 3\right )}{x + e^{\left (2 \, x\right )} + 3} \]

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

Mupad [B] (verification not implemented)

Time = 13.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\ln \left (x\right )-\ln \left (x+\frac {3}{82}\right )-\frac {\frac {{\mathrm {e}}^{2\,x+1}}{3}+\frac {x\,\mathrm {e}}{3}}{x+{\mathrm {e}}^{2\,x}+3} \]

[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)