3.27.33 \(\int \frac {-1+2 x+e^{-2+e^4} (-20+40 x)+e^{-4+2 e^4} (-158+313 x+4 x^2)+e^{-6+3 e^4} (-580+1130 x+40 x^2)+e^{-8+4 e^4} (-841+1595 x+115 x^2+2 x^3)}{x+20 e^{-2+e^4} x+e^{-4+2 e^4} (158 x+2 x^2)+e^{-6+3 e^4} (580 x+20 x^2)+e^{-8+4 e^4} (841 x+58 x^2+x^3)} \, dx\)

Optimal. Leaf size=30 \[ 7+2 x-\frac {x}{4+\left (5+e^{2-e^4}\right )^2+x}-\log (x) \]

________________________________________________________________________________________

Rubi [B]  time = 0.22, antiderivative size = 64, normalized size of antiderivative = 2.13, number of steps used = 3, number of rules used = 2, integrand size = 158, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6, 2074} \begin {gather*} 2 x+\frac {e^4+29 e^{2 e^4}+10 e^{2+e^4}}{e^{2 e^4} x+10 e^{2+e^4}+29 e^{2 e^4}+e^4}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x + E^(-2 + E^4)*(-20 + 40*x) + E^(-4 + 2*E^4)*(-158 + 313*x + 4*x^2) + E^(-6 + 3*E^4)*(-580 + 113
0*x + 40*x^2) + E^(-8 + 4*E^4)*(-841 + 1595*x + 115*x^2 + 2*x^3))/(x + 20*E^(-2 + E^4)*x + E^(-4 + 2*E^4)*(158
*x + 2*x^2) + E^(-6 + 3*E^4)*(580*x + 20*x^2) + E^(-8 + 4*E^4)*(841*x + 58*x^2 + x^3)),x]

[Out]

2*x + (E^4 + 29*E^(2*E^4) + 10*E^(2 + E^4))/(E^4 + 29*E^(2*E^4) + 10*E^(2 + E^4) + E^(2*E^4)*x) - Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+2 x+e^{-2+e^4} (-20+40 x)+e^{-4+2 e^4} \left (-158+313 x+4 x^2\right )+e^{-6+3 e^4} \left (-580+1130 x+40 x^2\right )+e^{-8+4 e^4} \left (-841+1595 x+115 x^2+2 x^3\right )}{\left (1+20 e^{-2+e^4}\right ) x+e^{-4+2 e^4} \left (158 x+2 x^2\right )+e^{-6+3 e^4} \left (580 x+20 x^2\right )+e^{-8+4 e^4} \left (841 x+58 x^2+x^3\right )} \, dx\\ &=\int \left (2-\frac {1}{x}-\frac {e^{2 e^4} \left (e^4+29 e^{2 e^4}+10 e^{2+e^4}\right )}{\left (e^4+29 e^{2 e^4}+10 e^{2+e^4}+e^{2 e^4} x\right )^2}\right ) \, dx\\ &=2 x+\frac {e^4+29 e^{2 e^4}+10 e^{2+e^4}}{e^4+29 e^{2 e^4}+10 e^{2+e^4}+e^{2 e^4} x}-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.15, size = 97, normalized size = 3.23 \begin {gather*} \frac {e^4 (1+2 x)+10 e^{2+e^4} (1+2 x)+e^{2 e^4} \left (29+58 x+2 x^2\right )-\left (e^4+10 e^{2+e^4}+e^{2 e^4} (29+x)\right ) \log (x)}{e^4+10 e^{2+e^4}+e^{2 e^4} (29+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x + E^(-2 + E^4)*(-20 + 40*x) + E^(-4 + 2*E^4)*(-158 + 313*x + 4*x^2) + E^(-6 + 3*E^4)*(-580
+ 1130*x + 40*x^2) + E^(-8 + 4*E^4)*(-841 + 1595*x + 115*x^2 + 2*x^3))/(x + 20*E^(-2 + E^4)*x + E^(-4 + 2*E^4
)*(158*x + 2*x^2) + E^(-6 + 3*E^4)*(580*x + 20*x^2) + E^(-8 + 4*E^4)*(841*x + 58*x^2 + x^3)),x]

[Out]

(E^4*(1 + 2*x) + 10*E^(2 + E^4)*(1 + 2*x) + E^(2*E^4)*(29 + 58*x + 2*x^2) - (E^4 + 10*E^(2 + E^4) + E^(2*E^4)*
(29 + x))*Log[x])/(E^4 + 10*E^(2 + E^4) + E^(2*E^4)*(29 + x))

________________________________________________________________________________________

fricas [B]  time = 0.72, size = 82, normalized size = 2.73 \begin {gather*} \frac {{\left (2 \, x^{2} + 58 \, x + 29\right )} e^{\left (2 \, e^{4} - 4\right )} + 10 \, {\left (2 \, x + 1\right )} e^{\left (e^{4} - 2\right )} - {\left ({\left (x + 29\right )} e^{\left (2 \, e^{4} - 4\right )} + 10 \, e^{\left (e^{4} - 2\right )} + 1\right )} \log \relax (x) + 2 \, x + 1}{{\left (x + 29\right )} e^{\left (2 \, e^{4} - 4\right )} + 10 \, e^{\left (e^{4} - 2\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+115*x^2+1595*x-841)*exp(exp(4)-2)^4+(40*x^2+1130*x-580)*exp(exp(4)-2)^3+(4*x^2+313*x-158)*ex
p(exp(4)-2)^2+(40*x-20)*exp(exp(4)-2)+2*x-1)/((x^3+58*x^2+841*x)*exp(exp(4)-2)^4+(20*x^2+580*x)*exp(exp(4)-2)^
3+(2*x^2+158*x)*exp(exp(4)-2)^2+20*x*exp(exp(4)-2)+x),x, algorithm="fricas")

[Out]

((2*x^2 + 58*x + 29)*e^(2*e^4 - 4) + 10*(2*x + 1)*e^(e^4 - 2) - ((x + 29)*e^(2*e^4 - 4) + 10*e^(e^4 - 2) + 1)*
log(x) + 2*x + 1)/((x + 29)*e^(2*e^4 - 4) + 10*e^(e^4 - 2) + 1)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+115*x^2+1595*x-841)*exp(exp(4)-2)^4+(40*x^2+1130*x-580)*exp(exp(4)-2)^3+(4*x^2+313*x-158)*ex
p(exp(4)-2)^2+(40*x-20)*exp(exp(4)-2)+2*x-1)/((x^3+58*x^2+841*x)*exp(exp(4)-2)^4+(20*x^2+580*x)*exp(exp(4)-2)^
3+(2*x^2+158*x)*exp(exp(4)-2)^2+20*x*exp(exp(4)-2)+x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -ln(abs(sageVARx))+(-58*exp(4*exp(4)-8)-
20*exp(3*exp(4)-6)-2*exp(2*exp(4)-4))*1/2/sqrt(-exp(2*exp(4)-4)^2-20*exp(2*exp(4)-4)*exp(3*exp(4)-6)+100*exp(2
*exp(4)-4)*exp(4*exp(

________________________________________________________________________________________

maple [B]  time = 0.23, size = 104, normalized size = 3.47

method	result	size
norman	\(\frac {2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{-4} x^{2}-{\mathrm e}^{-2 \,{\mathrm e}^{4}} {\mathrm e}^{4} \left (1653 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} {\mathrm e}^{-8}+1150 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} {\mathrm e}^{-6}+315 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{-4}+40 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}^{-2}+2\right )}{{\mathrm e}^{2 \,{\mathrm e}^{4}-4} x +29 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-4}+10 \,{\mathrm e}^{{\mathrm e}^{4}-2}+1}-\ln \relax (x )\)	\(104\)
risch	\(2 x +\frac {29 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-4}}{{\mathrm e}^{2 \,{\mathrm e}^{4}-4} x +29 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-4}+10 \,{\mathrm e}^{{\mathrm e}^{4}-2}+1}+\frac {10 \,{\mathrm e}^{{\mathrm e}^{4}-2}}{{\mathrm e}^{2 \,{\mathrm e}^{4}-4} x +29 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-4}+10 \,{\mathrm e}^{{\mathrm e}^{4}-2}+1}+\frac {1}{{\mathrm e}^{2 \,{\mathrm e}^{4}-4} x +29 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-4}+10 \,{\mathrm e}^{{\mathrm e}^{4}-2}+1}-\ln \relax (x )\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+115*x^2+1595*x-841)*exp(exp(4)-2)^4+(40*x^2+1130*x-580)*exp(exp(4)-2)^3+(4*x^2+313*x-158)*exp(exp(
4)-2)^2+(40*x-20)*exp(exp(4)-2)+2*x-1)/((x^3+58*x^2+841*x)*exp(exp(4)-2)^4+(20*x^2+580*x)*exp(exp(4)-2)^3+(2*x
^2+158*x)*exp(exp(4)-2)^2+20*x*exp(exp(4)-2)+x),x,method=_RETURNVERBOSE)

[Out]

(2*exp(exp(4))^2*exp(-2)^2*x^2-1/exp(exp(4))^2/exp(-2)^2*(1653*exp(exp(4))^4*exp(-2)^4+1150*exp(exp(4))^3*exp(
-2)^3+315*exp(exp(4))^2*exp(-2)^2+40*exp(exp(4))*exp(-2)+2))/(exp(exp(4)-2)^2*x+29*exp(exp(4)-2)^2+10*exp(exp(
4)-2)+1)-ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 52, normalized size = 1.73 \begin {gather*} 2 \, x + \frac {e^{4} + 29 \, e^{\left (2 \, e^{4}\right )} + 10 \, e^{\left (e^{4} + 2\right )}}{x e^{\left (2 \, e^{4}\right )} + e^{4} + 29 \, e^{\left (2 \, e^{4}\right )} + 10 \, e^{\left (e^{4} + 2\right )}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+115*x^2+1595*x-841)*exp(exp(4)-2)^4+(40*x^2+1130*x-580)*exp(exp(4)-2)^3+(4*x^2+313*x-158)*ex
p(exp(4)-2)^2+(40*x-20)*exp(exp(4)-2)+2*x-1)/((x^3+58*x^2+841*x)*exp(exp(4)-2)^4+(20*x^2+580*x)*exp(exp(4)-2)^
3+(2*x^2+158*x)*exp(exp(4)-2)^2+20*x*exp(exp(4)-2)+x),x, algorithm="maxima")

[Out]

2*x + (e^4 + 29*e^(2*e^4) + 10*e^(e^4 + 2))/(x*e^(2*e^4) + e^4 + 29*e^(2*e^4) + 10*e^(e^4 + 2)) - log(x)

________________________________________________________________________________________

mupad [B]  time = 1.81, size = 48, normalized size = 1.60 \begin {gather*} 2\,x-\ln \relax (x)+\frac {10\,{\mathrm {e}}^{2-{\mathrm {e}}^4}+{\mathrm {e}}^{4-2\,{\mathrm {e}}^4}+29}{x+10\,{\mathrm {e}}^{2-{\mathrm {e}}^4}+{\mathrm {e}}^{4-2\,{\mathrm {e}}^4}+29} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(2*exp(4) - 4)*(313*x + 4*x^2 - 158) + exp(3*exp(4) - 6)*(1130*x + 40*x^2 - 580) + exp(4*exp(4)
- 8)*(1595*x + 115*x^2 + 2*x^3 - 841) + exp(exp(4) - 2)*(40*x - 20) - 1)/(x + 20*x*exp(exp(4) - 2) + exp(4*exp
(4) - 8)*(841*x + 58*x^2 + x^3) + exp(2*exp(4) - 4)*(158*x + 2*x^2) + exp(3*exp(4) - 6)*(580*x + 20*x^2)),x)

[Out]

2*x - log(x) + (10*exp(2 - exp(4)) + exp(4 - 2*exp(4)) + 29)/(x + 10*exp(2 - exp(4)) + exp(4 - 2*exp(4)) + 29)

________________________________________________________________________________________

sympy [B]  time = 1.37, size = 58, normalized size = 1.93 \begin {gather*} 2 x - \log {\relax (x )} + \frac {e^{4} + 10 e^{2} e^{e^{4}} + 29 e^{2 e^{4}}}{x e^{2 e^{4}} + e^{4} + 10 e^{2} e^{e^{4}} + 29 e^{2 e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+115*x**2+1595*x-841)*exp(exp(4)-2)**4+(40*x**2+1130*x-580)*exp(exp(4)-2)**3+(4*x**2+313*x-1
58)*exp(exp(4)-2)**2+(40*x-20)*exp(exp(4)-2)+2*x-1)/((x**3+58*x**2+841*x)*exp(exp(4)-2)**4+(20*x**2+580*x)*exp
(exp(4)-2)**3+(2*x**2+158*x)*exp(exp(4)-2)**2+20*x*exp(exp(4)-2)+x),x)

[Out]

2*x - log(x) + (exp(4) + 10*exp(2)*exp(exp(4)) + 29*exp(2*exp(4)))/(x*exp(2*exp(4)) + exp(4) + 10*exp(2)*exp(e
xp(4)) + 29*exp(2*exp(4)))

________________________________________________________________________________________