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3.30.54 \(\int \frac {-5 x-7 x^2-4 x^3+e^5 (-5-2 x-2 x^2)+e^x (-x-x^2-x^3+e^5 (-1-x^2))+(-e^5-x-x^2) \log (5)+(10 x+6 x^2+e^5 (5+4 x)+e^x (2 x+x^2+e^5 (1+x))+(e^5+2 x) \log (5)) \log (x)}{e^{2 x} (e^{10} x^2+2 e^5 x^3+x^4) \log (5)+(25 x^4+20 x^5+4 x^6+e^{10} (25 x^2+20 x^3+4 x^4)+e^5 (50 x^3+40 x^4+8 x^5)) \log (5)+(10 x^4+4 x^5+e^{10} (10 x^2+4 x^3)+e^5 (20 x^3+8 x^4)) \log ^2(5)+(e^{10} x^2+2 e^5 x^3+x^4) \log ^3(5)+e^x ((10 x^4+4 x^5+e^{10} (10 x^2+4 x^3)+e^5 (20 x^3+8 x^4)) \log (5)+(2 e^{10} x^2+4 e^5 x^3+2 x^4) \log ^2(5))} \, dx\)

Optimal. Leaf size=33 \[ \frac {x-\log (x)}{x \left (e^5+x\right ) \log (5) \left (5+e^x+2 x+\log (5)\right )} \]

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Rubi [F]  time = 18.08, 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 {gather*} \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x - 7*x^2 - 4*x^3 + E^5*(-5 - 2*x - 2*x^2) + E^x*(-x - x^2 - x^3 + E^5*(-1 - x^2)) + (-E^5 - x - x^2)*
Log[5] + (10*x + 6*x^2 + E^5*(5 + 4*x) + E^x*(2*x + x^2 + E^5*(1 + x)) + (E^5 + 2*x)*Log[5])*Log[x])/(E^(2*x)*
(E^10*x^2 + 2*E^5*x^3 + x^4)*Log[5] + (25*x^4 + 20*x^5 + 4*x^6 + E^10*(25*x^2 + 20*x^3 + 4*x^4) + E^5*(50*x^3
+ 40*x^4 + 8*x^5))*Log[5] + (10*x^4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5]^2 + (E^10*x
^2 + 2*E^5*x^3 + x^4)*Log[5]^3 + E^x*((10*x^4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5] +
 (2*E^10*x^2 + 4*E^5*x^3 + 2*x^4)*Log[5]^2)),x]

[Out]

(2*Defer[Int][(-E^x - 2*x - 5*(1 + Log[5]/5))^(-1), x])/(E^15*Log[5]) + (2*Defer[Int][(-E^x - 2*x - 5*(1 + Log
[5]/5))^(-1), x])/(E^5*Log[5]) + (2*(1 + E^5)*Defer[Int][(-E^x - 2*x - 5*(1 + Log[5]/5))^(-1), x])/(E^10*Log[5
]) + (2*Log[x]*Defer[Int][(-E^x - 2*x - 5*(1 + Log[5]/5))^(-1), x])/(E^10*Log[5]) + Defer[Int][1/(x^2*(-E^x -
2*x - 5*(1 + Log[5]/5))), x]/(E^5*Log[5]) + Defer[Int][1/(x*(-E^x - 2*x - 5*(1 + Log[5]/5))), x]/(E^10*Log[5])
 + (2*Log[x]*Defer[Int][1/(x*(-E^x - 2*x - 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + (2*Defer[Int][x/(-E^x - 2*x
 - 5*(1 + Log[5]/5)), x])/(E^10*Log[5]) + (2*(1 + E^5)*Defer[Int][x/(-E^x - 2*x - 5*(1 + Log[5]/5)), x])/(E^15
*Log[5]) + (2*Log[x]*Defer[Int][x/(-E^x - 2*x - 5*(1 + Log[5]/5)), x])/(E^15*Log[5]) + (2*Defer[Int][x^2/(-E^x
 - 2*x - 5*(1 + Log[5]/5)), x])/(E^15*Log[5]) + Defer[Int][1/((E^5 + x)^2*(-E^x - 2*x - 5*(1 + Log[5]/5))), x]
/(E^5*Log[5]) + ((1 + E^5)*Defer[Int][1/((E^5 + x)^2*(-E^x - 2*x - 5*(1 + Log[5]/5))), x])/Log[5] + ((2 + E^5)
*Log[x]*Defer[Int][1/((E^5 + x)^2*(-E^x - 2*x - 5*(1 + Log[5]/5))), x])/(E^5*Log[5]) + ((3 + Log[5])*Defer[Int
][(E^x + 2*x + 5*(1 + Log[5]/5))^(-2), x])/(E^5*Log[5]) - ((3 - 2*E^5 + Log[5])*Defer[Int][(E^x + 2*x + 5*(1 +
 Log[5]/5))^(-2), x])/(E^5*Log[5]) - ((3 + Log[5])*Log[x]*Defer[Int][1/(x*(E^x + 2*x + 5*(1 + Log[5]/5))^2), x
])/(E^5*Log[5]) + ((3 - 2*E^5 + Log[5])*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))^2), x])/Log[5]
+ ((3 - 2*E^5 + Log[5])*Log[x]*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))^2), x])/(E^5*Log[5]) + (
2*Defer[Int][(E^x + 2*x + 5*(1 + Log[5]/5))^(-1), x])/(E^15*Log[5]) + (2*Defer[Int][(E^x + 2*x + 5*(1 + Log[5]
/5))^(-1), x])/(E^5*Log[5]) + (2*(1 + E^5)*Defer[Int][(E^x + 2*x + 5*(1 + Log[5]/5))^(-1), x])/(E^10*Log[5]) +
 (2*Log[x]*Defer[Int][(E^x + 2*x + 5*(1 + Log[5]/5))^(-1), x])/(E^10*Log[5]) + (3*Defer[Int][1/((-E^5 - x)*(E^
x + 2*x + 5*(1 + Log[5]/5))), x])/Log[5] + (2*(1 + E^(-5))*Defer[Int][1/((-E^5 - x)*(E^x + 2*x + 5*(1 + Log[5]
/5))), x])/Log[5] + (3*Defer[Int][1/((-E^5 - x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + (2*Log[x]
*Defer[Int][1/((-E^5 - x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^5*Log[5]) + (2*(2 + E^5)*Log[x]*Defer[Int][1
/((-E^5 - x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + (Log[x]*Defer[Int][1/(x^2*(E^x + 2*x + 5*(1
+ Log[5]/5))), x])/(E^5*Log[5]) + (2*Defer[Int][1/(x*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + ((2
+ E^5)*Log[x]*Defer[Int][1/(x*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + (2*Defer[Int][x/(E^x + 2*x
+ 5*(1 + Log[5]/5)), x])/(E^10*Log[5]) + (2*(1 + E^5)*Defer[Int][x/(E^x + 2*x + 5*(1 + Log[5]/5)), x])/(E^15*L
og[5]) + (2*Log[x]*Defer[Int][x/(E^x + 2*x + 5*(1 + Log[5]/5)), x])/(E^15*Log[5]) + (2*Defer[Int][x^2/(E^x + 2
*x + 5*(1 + Log[5]/5)), x])/(E^15*Log[5]) + Defer[Int][1/((E^5 + x)^2*(E^x + 2*x + 5*(1 + Log[5]/5))), x]/(E^5
*Log[5]) + (E^5*Defer[Int][1/((E^5 + x)^2*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/Log[5] + (Log[x]*Defer[Int][1/(
(E^5 + x)^2*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/Log[5] + (Log[x]*Defer[Int][1/((E^5 + x)^2*(E^x + 2*x + 5*(1
+ Log[5]/5))), x])/(E^5*Log[5]) + (2*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/Log[5] + (2*
(1 + E^(-5))*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/Log[5] + (2*Defer[Int][1/((E^5 + x)*
(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + (2*Log[x]*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log
[5]/5))), x])/(E^10*Log[5]) + (2*Log[x]*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^5*Log[
5]) + ((2 + E^5)*Log[x]*Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x])/(E^10*Log[5]) + ((3 + Log
[5])*Defer[Int][Defer[Int][1/(x*(E^x + 2*x + 5*(1 + Log[5]/5))^2), x]/x, x])/(E^5*Log[5]) - ((3 - 2*E^5 + Log[
5])*Defer[Int][Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))^2), x]/x, x])/(E^5*Log[5]) - (2*Defer[In
t][Defer[Int][-(E^x + 2*x + 5*(1 + Log[5]/5))^(-1), x]/x, x])/(E^10*Log[5]) - (2*Defer[Int][Defer[Int][(E^x +
2*x + 5*(1 + Log[5]/5))^(-1), x]/x, x])/(E^10*Log[5]) - Defer[Int][Defer[Int][1/(x^2*(E^x + 2*x + 5*(1 + Log[5
]/5))), x]/x, x]/(E^5*Log[5]) - (2*Defer[Int][Defer[Int][-(1/(x*(E^x + 2*x + 5*(1 + Log[5]/5)))), x]/x, x])/(E
^10*Log[5]) - ((2 + E^5)*Defer[Int][Defer[Int][1/(x*(E^x + 2*x + 5*(1 + Log[5]/5))), x]/x, x])/(E^10*Log[5]) -
 (2*Defer[Int][Defer[Int][-(x/(E^x + 2*x + 5*(1 + Log[5]/5))), x]/x, x])/(E^15*Log[5]) - (2*Defer[Int][Defer[I
nt][x/(E^x + 2*x + 5*(1 + Log[5]/5)), x]/x, x])/(E^15*Log[5]) - ((2 + E^5)*Defer[Int][Defer[Int][-(1/((E^5 + x
)^2*(E^x + 2*x + 5*(1 + Log[5]/5)))), x]/x, x])/(E^5*Log[5]) - Defer[Int][Defer[Int][1/((E^5 + x)^2*(E^x + 2*x
 + 5*(1 + Log[5]/5))), x]/x, x]/Log[5] - Defer[Int][Defer[Int][1/((E^5 + x)^2*(E^x + 2*x + 5*(1 + Log[5]/5))),
 x]/x, x]/(E^5*Log[5]) - (2*Defer[Int][Defer[Int][-(1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5)))), x]/x, x])/(
E^5*Log[5]) - (2*(2 + E^5)*Defer[Int][Defer[Int][-(1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5)))), x]/x, x])/(E
^10*Log[5]) - (2*Defer[Int][Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x]/x, x])/(E^10*Log[5]) -
 (2*Defer[Int][Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x]/x, x])/(E^5*Log[5]) - ((2 + E^5)*De
fer[Int][Defer[Int][1/((E^5 + x)*(E^x + 2*x + 5*(1 + Log[5]/5))), x]/x, x])/(E^10*Log[5])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{5+x} \left (1+x^2\right )-e^x x \left (1+x+x^2\right )-e^5 \left (5+2 x+2 x^2+\log (5)\right )-x \left (5+4 x^2+\log (5)+x (7+\log (5))\right )+\left (e^{5+x} (1+x)+e^x x (2+x)+2 x (5+3 x+\log (5))+e^5 (5+4 x+\log (5))\right ) \log (x)}{x^2 \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2 \log (5)} \, dx\\ &=\frac {\int \frac {-e^{5+x} \left (1+x^2\right )-e^x x \left (1+x+x^2\right )-e^5 \left (5+2 x+2 x^2+\log (5)\right )-x \left (5+4 x^2+\log (5)+x (7+\log (5))\right )+\left (e^{5+x} (1+x)+e^x x (2+x)+2 x (5+3 x+\log (5))+e^5 (5+4 x+\log (5))\right ) \log (x)}{x^2 \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{\log (5)}\\ &=\frac {\int \left (\frac {(3+2 x+\log (5)) (x-\log (x))}{x \left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}+\frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{x^2 \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}\right ) \, dx}{\log (5)}\\ &=\frac {\int \frac {(3+2 x+\log (5)) (x-\log (x))}{x \left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{\log (5)}+\frac {\int \frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{x^2 \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{\log (5)}\\ &=\frac {\int \frac {-e^5 \left (1+x^2\right )-x \left (1+x+x^2\right )+\left (e^5 (1+x)+x (2+x)\right ) \log (x)}{x^2 \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{\log (5)}+\frac {\int \left (\frac {(3+\log (5)) (x-\log (x))}{e^5 x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}+\frac {\left (3-2 e^5+\log (5)\right ) (-x+\log (x))}{e^5 \left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}\right ) \, dx}{\log (5)}\\ &=\frac {\int \left (\frac {2 \left (e^5+x+\left (1+e^5\right ) x^2+x^3-e^5 \log (x)-2 \left (1+\frac {e^5}{2}\right ) x \log (x)-x^2 \log (x)\right )}{e^{15} x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}+\frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{e^{10} x^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}+\frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{e^{10} \left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}+\frac {2 \left (-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)\right )}{e^{15} \left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}\right ) \, dx}{\log (5)}+\frac {(3+\log (5)) \int \frac {x-\log (x)}{x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}+\frac {\left (3-2 e^5+\log (5)\right ) \int \frac {-x+\log (x)}{\left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}\\ &=\frac {2 \int \frac {e^5+x+\left (1+e^5\right ) x^2+x^3-e^5 \log (x)-2 \left (1+\frac {e^5}{2}\right ) x \log (x)-x^2 \log (x)}{x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{15} \log (5)}+\frac {2 \int \frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{\left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{15} \log (5)}+\frac {\int \frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{x^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{10} \log (5)}+\frac {\int \frac {-e^5-x-\left (1+e^5\right ) x^2-x^3+e^5 \log (x)+2 \left (1+\frac {e^5}{2}\right ) x \log (x)+x^2 \log (x)}{\left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{10} \log (5)}+\frac {(3+\log (5)) \int \left (\frac {1}{\left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}-\frac {\log (x)}{x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}\right ) \, dx}{e^5 \log (5)}+\frac {\left (3-2 e^5+\log (5)\right ) \int \left (\frac {x}{\left (-e^5-x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}+\frac {\log (x)}{\left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}\right ) \, dx}{e^5 \log (5)}\\ &=\frac {2 \int \frac {-e^5 \left (1+x^2\right )-x \left (1+x+x^2\right )+\left (e^5 (1+x)+x (2+x)\right ) \log (x)}{\left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{15} \log (5)}+\frac {2 \int \left (\frac {1}{e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )}+\frac {e^5}{x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )}+\frac {\left (1+e^5\right ) x}{e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )}+\frac {x^2}{e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )}+\frac {e^5 \log (x)}{x \left (-e^x-2 x-5 \left (1+\frac {\log (5)}{5}\right )\right )}+\frac {x \log (x)}{-e^x-2 x-5 \left (1+\frac {\log (5)}{5}\right )}+\frac {\left (-2-e^5\right ) \log (x)}{e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )}\right ) \, dx}{e^{15} \log (5)}+\frac {\int \frac {-e^5 \left (1+x^2\right )-x \left (1+x+x^2\right )+\left (e^5 (1+x)+x (2+x)\right ) \log (x)}{x^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{10} \log (5)}+\frac {\int \frac {-e^5 \left (1+x^2\right )-x \left (1+x+x^2\right )+\left (e^5 (1+x)+x (2+x)\right ) \log (x)}{\left (e^5+x\right )^2 \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )} \, dx}{e^{10} \log (5)}+\frac {(3+\log (5)) \int \frac {1}{\left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}-\frac {(3+\log (5)) \int \frac {\log (x)}{x \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}+\frac {\left (3-2 e^5+\log (5)\right ) \int \frac {x}{\left (-e^5-x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}+\frac {\left (3-2 e^5+\log (5)\right ) \int \frac {\log (x)}{\left (e^5+x\right ) \left (e^x+2 x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2} \, dx}{e^5 \log (5)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 34, normalized size = 1.03 \begin {gather*} -\frac {-x+\log (x)}{x \left (e^5+x\right ) \log (5) \left (5+e^x+2 x+\log (5)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x - 7*x^2 - 4*x^3 + E^5*(-5 - 2*x - 2*x^2) + E^x*(-x - x^2 - x^3 + E^5*(-1 - x^2)) + (-E^5 - x -
 x^2)*Log[5] + (10*x + 6*x^2 + E^5*(5 + 4*x) + E^x*(2*x + x^2 + E^5*(1 + x)) + (E^5 + 2*x)*Log[5])*Log[x])/(E^
(2*x)*(E^10*x^2 + 2*E^5*x^3 + x^4)*Log[5] + (25*x^4 + 20*x^5 + 4*x^6 + E^10*(25*x^2 + 20*x^3 + 4*x^4) + E^5*(5
0*x^3 + 40*x^4 + 8*x^5))*Log[5] + (10*x^4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5]^2 + (
E^10*x^2 + 2*E^5*x^3 + x^4)*Log[5]^3 + E^x*((10*x^4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Lo
g[5] + (2*E^10*x^2 + 4*E^5*x^3 + 2*x^4)*Log[5]^2)),x]

[Out]

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

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fricas [A]  time = 1.25, size = 62, normalized size = 1.88 \begin {gather*} \frac {x - \log \relax (x)}{{\left (x^{2} + x e^{5}\right )} e^{x} \log \relax (5) + {\left (x^{2} + x e^{5}\right )} \log \relax (5)^{2} + {\left (2 \, x^{3} + 5 \, x^{2} + {\left (2 \, x^{2} + 5 \, x\right )} e^{5}\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(4*x+5)*exp(5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(
5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*log(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)
+x^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*ex
p(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*exp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*
x^3)*exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x^4+50*x^3)*exp(5)+4*x^6+20*x^5+2
5*x^4)*log(5)),x, algorithm="fricas")

[Out]

(x - log(x))/((x^2 + x*e^5)*e^x*log(5) + (x^2 + x*e^5)*log(5)^2 + (2*x^3 + 5*x^2 + (2*x^2 + 5*x)*e^5)*log(5))

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giac [B]  time = 0.74, size = 72, normalized size = 2.18 \begin {gather*} \frac {x - \log \relax (x)}{2 \, x^{3} \log \relax (5) + 2 \, x^{2} e^{5} \log \relax (5) + x^{2} e^{x} \log \relax (5) + x^{2} \log \relax (5)^{2} + x e^{5} \log \relax (5)^{2} + 5 \, x^{2} \log \relax (5) + 5 \, x e^{5} \log \relax (5) + x e^{\left (x + 5\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(4*x+5)*exp(5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(
5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*log(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)
+x^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*ex
p(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*exp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*
x^3)*exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x^4+50*x^3)*exp(5)+4*x^6+20*x^5+2
5*x^4)*log(5)),x, algorithm="giac")

[Out]

(x - log(x))/(2*x^3*log(5) + 2*x^2*e^5*log(5) + x^2*e^x*log(5) + x^2*log(5)^2 + x*e^5*log(5)^2 + 5*x^2*log(5)
+ 5*x*e^5*log(5) + x*e^(x + 5)*log(5))

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maple [A]  time = 0.08, size = 52, normalized size = 1.58

method result size
risch \(-\frac {\ln \relax (x )}{x \ln \relax (5) \left ({\mathrm e}^{5}+x \right ) \left (2 x +{\mathrm e}^{x}+\ln \relax (5)+5\right )}+\frac {1}{\ln \relax (5) \left ({\mathrm e}^{5}+x \right ) \left (2 x +{\mathrm e}^{x}+\ln \relax (5)+5\right )}\) \(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x+1)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*ln(5)+(4*x+5)*exp(5)+6*x^2+10*x)*ln(x)+((-x^2-1)*exp(5)-x^3-x
^2-x)*exp(x)+(-exp(5)-x^2-x)*ln(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)+x^4)*ln(
5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*ln(5)^2+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5+
10*x^4)*ln(5))*exp(x)+(x^2*exp(5)^2+2*x^3*exp(5)+x^4)*ln(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4
*x^5+10*x^4)*ln(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x^4+50*x^3)*exp(5)+4*x^6+20*x^5+25*x^4)*ln(5)),
x,method=_RETURNVERBOSE)

[Out]

-1/x/ln(5)/(exp(5)+x)/(2*x+exp(x)+ln(5)+5)*ln(x)+1/ln(5)/(exp(5)+x)/(2*x+exp(x)+ln(5)+5)

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maxima [B]  time = 0.90, size = 65, normalized size = 1.97 \begin {gather*} \frac {x - \log \relax (x)}{2 \, x^{3} \log \relax (5) + {\left (2 \, e^{5} \log \relax (5) + \log \relax (5)^{2} + 5 \, \log \relax (5)\right )} x^{2} + {\left (\log \relax (5)^{2} + 5 \, \log \relax (5)\right )} x e^{5} + {\left (x^{2} \log \relax (5) + x e^{5} \log \relax (5)\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(4*x+5)*exp(5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(
5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*log(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)
+x^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*ex
p(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*exp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*
x^3)*exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x^4+50*x^3)*exp(5)+4*x^6+20*x^5+2
5*x^4)*log(5)),x, algorithm="maxima")

[Out]

(x - log(x))/(2*x^3*log(5) + (2*e^5*log(5) + log(5)^2 + 5*log(5))*x^2 + (log(5)^2 + 5*log(5))*x*e^5 + (x^2*log
(5) + x*e^5*log(5))*e^x)

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mupad [B]  time = 3.57, size = 31, normalized size = 0.94 \begin {gather*} \frac {x-\ln \relax (x)}{x\,\ln \relax (5)\,\left (x+{\mathrm {e}}^5\right )\,\left (2\,x+\ln \relax (5)+{\mathrm {e}}^x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + exp(5)*(2*x + 2*x^2 + 5) + exp(x)*(x + x^2 + x^3 + exp(5)*(x^2 + 1)) + 7*x^2 + 4*x^3 + log(5)*(x +
 exp(5) + x^2) - log(x)*(10*x + log(5)*(2*x + exp(5)) + 6*x^2 + exp(x)*(2*x + exp(5)*(x + 1) + x^2) + exp(5)*(
4*x + 5)))/(log(5)^2*(exp(10)*(10*x^2 + 4*x^3) + exp(5)*(20*x^3 + 8*x^4) + 10*x^4 + 4*x^5) + exp(x)*(log(5)*(e
xp(10)*(10*x^2 + 4*x^3) + exp(5)*(20*x^3 + 8*x^4) + 10*x^4 + 4*x^5) + log(5)^2*(4*x^3*exp(5) + 2*x^2*exp(10) +
 2*x^4)) + log(5)*(exp(10)*(25*x^2 + 20*x^3 + 4*x^4) + exp(5)*(50*x^3 + 40*x^4 + 8*x^5) + 25*x^4 + 20*x^5 + 4*
x^6) + log(5)^3*(2*x^3*exp(5) + x^2*exp(10) + x^4) + exp(2*x)*log(5)*(2*x^3*exp(5) + x^2*exp(10) + x^4)),x)

[Out]

(x - log(x))/(x*log(5)*(x + exp(5))*(2*x + log(5) + exp(x) + 5))

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sympy [B]  time = 0.52, size = 80, normalized size = 2.42 \begin {gather*} \frac {x - \log {\relax (x )}}{2 x^{3} \log {\relax (5 )} + x^{2} \log {\relax (5 )}^{2} + 5 x^{2} \log {\relax (5 )} + 2 x^{2} e^{5} \log {\relax (5 )} + x e^{5} \log {\relax (5 )}^{2} + 5 x e^{5} \log {\relax (5 )} + \left (x^{2} \log {\relax (5 )} + x e^{5} \log {\relax (5 )}\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(5)+x**2+2*x)*exp(x)+(2*x+exp(5))*ln(5)+(4*x+5)*exp(5)+6*x**2+10*x)*ln(x)+((-x**2-1)*exp
(5)-x**3-x**2-x)*exp(x)+(-exp(5)-x**2-x)*ln(5)+(-2*x**2-2*x-5)*exp(5)-4*x**3-7*x**2-5*x)/((x**2*exp(5)**2+2*x*
*3*exp(5)+x**4)*ln(5)*exp(x)**2+((2*x**2*exp(5)**2+4*x**3*exp(5)+2*x**4)*ln(5)**2+((4*x**3+10*x**2)*exp(5)**2+
(8*x**4+20*x**3)*exp(5)+4*x**5+10*x**4)*ln(5))*exp(x)+(x**2*exp(5)**2+2*x**3*exp(5)+x**4)*ln(5)**3+((4*x**3+10
*x**2)*exp(5)**2+(8*x**4+20*x**3)*exp(5)+4*x**5+10*x**4)*ln(5)**2+((4*x**4+20*x**3+25*x**2)*exp(5)**2+(8*x**5+
40*x**4+50*x**3)*exp(5)+4*x**6+20*x**5+25*x**4)*ln(5)),x)

[Out]

(x - log(x))/(2*x**3*log(5) + x**2*log(5)**2 + 5*x**2*log(5) + 2*x**2*exp(5)*log(5) + x*exp(5)*log(5)**2 + 5*x
*exp(5)*log(5) + (x**2*log(5) + x*exp(5)*log(5))*exp(x))

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