\(\int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} (-1+2 x+x^2+x^3)+e^x (-49-149 x+53 x^2+2 x^3+2 x^4)+(49 x+48 x^3+2 x^4+3 x^5+e^{2 x} (-x+2 x^2+3 x^3)+e^x (49 x+96 x^2-45 x^3+6 x^4)) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx\) [8646]

Optimal result

Integrand size = 148, antiderivative size = 26 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\left (1+x+\frac {49}{e^x+x}\right ) \left (3+x \left (-\frac {1}{x}+x\right ) \log (x)\right ) \]

[Out]

(ln(x)*x*(x-1/x)+3)*(x+49/(exp(x)+x)+1)

Rubi [F]

\[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx \]

[In]

Int[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x^3) + E^x*(-49 - 149*x + 53*x^2 + 2*x^3 +
2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6*x^4))
*Log[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]

[Out]

3*x + 147/(E^x + x) - Log[x] - x*Log[x] + x^2*Log[x] + x^3*Log[x] - (49*Log[x])/(E^x + x) - 49*Log[x]*Defer[In
t][x^2/(E^x + x)^2, x] + 49*Log[x]*Defer[Int][x^3/(E^x + x)^2, x] + 49*Defer[Int][x/(E^x + x), x] + 98*Log[x]*
Defer[Int][x/(E^x + x), x] - 49*Log[x]*Defer[Int][x^2/(E^x + x), x] + 49*Defer[Int][Defer[Int][x^2/(E^x + x)^2
, x]/x, x] - 49*Defer[Int][Defer[Int][x^3/(E^x + x)^2, x]/x, x] - 98*Defer[Int][Defer[Int][x/(E^x + x), x]/x,
x] + 49*Defer[Int][Defer[Int][x^2/(E^x + x), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{x \left (e^x+x\right )^2} \, dx \\ & = \int \left (\frac {49 (-1+x) \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2}-\frac {49 \left (1+3 x-x^2-x \log (x)-2 x^2 \log (x)+x^3 \log (x)\right )}{x \left (e^x+x\right )}+\frac {-1+2 x+x^2+x^3-x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)}{x}\right ) \, dx \\ & = 49 \int \frac {(-1+x) \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2} \, dx-49 \int \frac {1+3 x-x^2-x \log (x)-2 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right )} \, dx+\int \frac {-1+2 x+x^2+x^3-x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)}{x} \, dx \\ & = -\left (49 \int \left (\frac {3}{e^x+x}+\frac {1}{x \left (e^x+x\right )}-\frac {x}{e^x+x}-\frac {\log (x)}{e^x+x}-\frac {2 x \log (x)}{e^x+x}+\frac {x^2 \log (x)}{e^x+x}\right ) \, dx\right )+49 \int \left (-\frac {3-\log (x)+x^2 \log (x)}{\left (e^x+x\right )^2}+\frac {x \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2}\right ) \, dx+\int \left (\frac {-1+2 x+x^2+x^3}{x}+\left (-1+2 x+3 x^2\right ) \log (x)\right ) \, dx \\ & = -\left (49 \int \frac {1}{x \left (e^x+x\right )} \, dx\right )+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\log (x)}{e^x+x} \, dx-49 \int \frac {x^2 \log (x)}{e^x+x} \, dx-49 \int \frac {3-\log (x)+x^2 \log (x)}{\left (e^x+x\right )^2} \, dx+49 \int \frac {x \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2} \, dx+98 \int \frac {x \log (x)}{e^x+x} \, dx-147 \int \frac {1}{e^x+x} \, dx+\int \frac {-1+2 x+x^2+x^3}{x} \, dx+\int \left (-1+2 x+3 x^2\right ) \log (x) \, dx \\ & = -\left (49 \int \frac {1}{x \left (e^x+x\right )} \, dx\right )+49 \int \frac {x}{e^x+x} \, dx-49 \int \left (\frac {3}{\left (e^x+x\right )^2}-\frac {\log (x)}{\left (e^x+x\right )^2}+\frac {x^2 \log (x)}{\left (e^x+x\right )^2}\right ) \, dx+49 \int \left (\frac {3 x}{\left (e^x+x\right )^2}-\frac {x \log (x)}{\left (e^x+x\right )^2}+\frac {x^3 \log (x)}{\left (e^x+x\right )^2}\right ) \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-147 \int \frac {1}{e^x+x} \, dx+(49 \log (x)) \int \frac {1}{e^x+x} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx+\int \left (2-\frac {1}{x}+x+x^2\right ) \, dx+\int \left (-\log (x)+2 x \log (x)+3 x^2 \log (x)\right ) \, dx \\ & = 2 x+\frac {x^2}{2}+\frac {x^3}{3}-\log (x)+2 \int x \log (x) \, dx+3 \int x^2 \log (x) \, dx-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {x \log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {x^2 \log (x)}{\left (e^x+x\right )^2} \, dx+49 \int \frac {x^3 \log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-147 \int \frac {1}{\left (e^x+x\right )^2} \, dx+147 \int \frac {x}{\left (e^x+x\right )^2} \, dx-147 \int \frac {1}{e^x+x} \, dx+(49 \log (x)) \int \frac {1}{e^x+x} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx-\int \log (x) \, dx \\ & = 3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\frac {1}{e^x+x}+\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx \\ & = 3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \left (\frac {1}{x \left (e^x+x\right )}+\frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x}\right ) \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx \\ & = 3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx \\ & = 3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \left (\frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x}+\frac {\int \frac {1}{e^x+x} \, dx}{x}\right ) \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx \\ & = 3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \left (49+e^x x+x^2\right )+\left (-1+x^2\right ) \left (49+x+x^2+e^x (1+x)\right ) \log (x)}{e^x+x} \]

[In]

Integrate[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x^3) + E^x*(-49 - 149*x + 53*x^2 + 2*
x^3 + 2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6
*x^4))*Log[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]

[Out]

(3*(49 + E^x*x + x^2) + (-1 + x^2)*(49 + x + x^2 + E^x*(1 + x))*Log[x])/(E^x + x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).

Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69

method	result	size
risch	\(\frac {\left (x^{4}+{\mathrm e}^{x} x^{3}+x^{3}+{\mathrm e}^{x} x^{2}+48 x^{2}-{\mathrm e}^{x} x -49\right ) \ln \left (x \right )}{{\mathrm e}^{x}+x}-\frac {x \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-3 x^{2}-3 \,{\mathrm e}^{x} x -147}{{\mathrm e}^{x}+x}\)	\(70\)
parallelrisch	\(-\frac {-x^{4} \ln \left (x \right )-x^{3} {\mathrm e}^{x} \ln \left (x \right )-x^{3} \ln \left (x \right )-x^{2} {\mathrm e}^{x} \ln \left (x \right )-48 x^{2} \ln \left (x \right )+x \,{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-3 x^{2}-3 \,{\mathrm e}^{x} x -147+49 \ln \left (x \right )}{{\mathrm e}^{x}+x}\)	\(79\)

[In]

int((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*ln(x)+(x^3+x^2+2*x-1
)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x,meth
od=_RETURNVERBOSE)

[Out]

(x^4+exp(x)*x^3+x^3+exp(x)*x^2+48*x^2-exp(x)*x-49)/(exp(x)+x)*ln(x)-(x*ln(x)+exp(x)*ln(x)-3*x^2-3*exp(x)*x-147
)/(exp(x)+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \, x^{2} + 3 \, x e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} + {\left (x^{3} + x^{2} - x - 1\right )} e^{x} - x - 49\right )} \log \left (x\right ) + 147}{x + e^{x}} \]

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="fricas")

[Out]

(3*x^2 + 3*x*e^x + (x^4 + x^3 + 48*x^2 + (x^3 + x^2 - x - 1)*e^x - x - 49)*log(x) + 147)/(x + e^x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=3 x + \left (x^{3} + x^{2} - x\right ) \log {\left (x \right )} - \log {\left (x \right )} + \frac {49 x^{2} \log {\left (x \right )} - 49 \log {\left (x \right )} + 147}{x + e^{x}} \]

[In]

integrate((((3*x**3+2*x**2-x)*exp(x)**2+(6*x**4-45*x**3+96*x**2+49*x)*exp(x)+3*x**5+2*x**4+48*x**3+49*x)*ln(x)
+(x**3+x**2+2*x-1)*exp(x)**2+(2*x**4+2*x**3+53*x**2-149*x-49)*exp(x)+x**5+x**4+51*x**3-x**2-196*x)/(x*exp(x)**
2+2*exp(x)*x**2+x**3),x)

[Out]

3*x + (x**3 + x**2 - x)*log(x) - log(x) + (49*x**2*log(x) - 49*log(x) + 147)/(x + exp(x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \, x^{2} + {\left ({\left (x^{3} + x^{2} - x - 1\right )} \log \left (x\right ) + 3 \, x\right )} e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} - x - 49\right )} \log \left (x\right ) + 147}{x + e^{x}} \]

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="maxima")

[Out]

(3*x^2 + ((x^3 + x^2 - x - 1)*log(x) + 3*x)*e^x + (x^4 + x^3 + 48*x^2 - x - 49)*log(x) + 147)/(x + e^x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {x^{4} \log \left (x\right ) + x^{3} e^{x} \log \left (x\right ) + x^{3} \log \left (x\right ) + x^{2} e^{x} \log \left (x\right ) + 48 \, x^{2} \log \left (x\right ) - x e^{x} \log \left (x\right ) + 3 \, x^{2} + 3 \, x e^{x} - x \log \left (x\right ) - e^{x} \log \left (x\right ) - 49 \, \log \left (x\right ) + 147}{x + e^{x}} \]

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="giac")

[Out]

(x^4*log(x) + x^3*e^x*log(x) + x^3*log(x) + x^2*e^x*log(x) + 48*x^2*log(x) - x*e^x*log(x) + 3*x^2 + 3*x*e^x -
x*log(x) - e^x*log(x) - 49*log(x) + 147)/(x + e^x)

Mupad [B] (verification not implemented)

Time = 13.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=3\,x-\ln \left (x\right )+\frac {147}{x+{\mathrm {e}}^x}+\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (x^3+x^2-x\right )+x\,\left (x^3+x^2-x\right )+49\,x^2-49\right )}{x+{\mathrm {e}}^x} \]

[In]

int((exp(x)*(53*x^2 - 149*x + 2*x^3 + 2*x^4 - 49) - 196*x + log(x)*(49*x + exp(x)*(49*x + 96*x^2 - 45*x^3 + 6*
x^4) + exp(2*x)*(2*x^2 - x + 3*x^3) + 48*x^3 + 2*x^4 + 3*x^5) + exp(2*x)*(2*x + x^2 + x^3 - 1) - x^2 + 51*x^3
+ x^4 + x^5)/(x*exp(2*x) + 2*x^2*exp(x) + x^3),x)

[Out]

3*x - log(x) + 147/(x + exp(x)) + (log(x)*(exp(x)*(x^2 - x + x^3) + x*(x^2 - x + x^3) + 49*x^2 - 49))/(x + exp
(x))