3.76.57 \(\int \frac {e^{\frac {x+(1-x-x^2) \log (e^{2 x}+2 e^{3 x}+e^{4 x}+x)}{\log (e^{2 x}+2 e^{3 x}+e^{4 x}+x)}} (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+(e^{2 x}+2 e^{3 x}+e^{4 x}+x) \log (e^{2 x}+2 e^{3 x}+e^{4 x}+x)+(e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2) \log ^2(e^{2 x}+2 e^{3 x}+e^{4 x}+x))}{(e^{2 x}+2 e^{3 x}+e^{4 x}+x) \log ^2(e^{2 x}+2 e^{3 x}+e^{4 x}+x)} \, dx\)

Optimal. Leaf size=30 \[ e^{1-x-x^2+\frac {x}{\log \left (\left (e^x+e^{2 x}\right )^2+x\right )}} \]

________________________________________________________________________________________

Rubi [F]  time = 21.74, 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 {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((x + (1 - x - x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])*(-x
- 2*E^(2*x)*x - 6*E^(3*x)*x - 4*E^(4*x)*x + (E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(
4*x) + x] + (E^(3*x)*(-2 - 4*x) + E^(2*x)*(-1 - 2*x) + E^(4*x)*(-1 - 2*x) - x - 2*x^2)*Log[E^(2*x) + 2*E^(3*x)
+ E^(4*x) + x]^2))/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2),x]

[Out]

-Defer[Int][E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]), x] - 2*Defer[Int][E^(1 - x - x^2 + x/L
og[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])*x, x] - 4*Defer[Int][(E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4
*x) + x])*x)/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2, x] - Defer[Int][(E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(
3*x) + E^(4*x) + x])*x)/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2), x] + 2
*Defer[Int][(E^(1 + x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])*x)/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x
)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2), x] + 2*Defer[Int][(E^(1 + 2*x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) +
E^(4*x) + x])*x)/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2), x] + 4*Defer
[Int][(E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])*x^2)/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Lo
g[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2), x] + Defer[Int][E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4*x)
+ x])/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-2 x \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx\\ &=\int \frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-2 x \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx\\ &=\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-(1+2 x) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx\\ &=\int \left (-\frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {2 e^{1+x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {2 \exp \left (1+2 x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {4 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x^2}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx+\int \left (-e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}-2 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x-\frac {4 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx\\ &=-\left (2 \int e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x \, dx\right )+2 \int \frac {e^{1+x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+2 \int \frac {\exp \left (1+2 x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx-4 \int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+4 \int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x^2}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx-\int e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \, dx-\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 36, normalized size = 1.20 \begin {gather*} e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x + (1 - x - x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x
])*(-x - 2*E^(2*x)*x - 6*E^(3*x)*x - 4*E^(4*x)*x + (E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x)
+ E^(4*x) + x] + (E^(3*x)*(-2 - 4*x) + E^(2*x)*(-1 - 2*x) + E^(4*x)*(-1 - 2*x) - x - 2*x^2)*Log[E^(2*x) + 2*E
^(3*x) + E^(4*x) + x]^2))/((E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2),x]

[Out]

E^(1 - x - x^2 + x/Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 50, normalized size = 1.67 \begin {gather*} e^{\left (-\frac {{\left (x^{2} + x - 1\right )} \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) - x}{\log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-2*x-1)*exp(x)^2-2*x^2-x)*log(exp(2*x)^2+2*exp(x)*ex
p(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*
x*exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+
x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*ex
p(x)*exp(2*x)+exp(x)^2+x)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-2*x-1)*exp(x)^2-2*x^2-x)*log(exp(2*x)^2+2*exp(x)*ex
p(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*
x*exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+
x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*ex
p(x)*exp(2*x)+exp(x)^2+x)^2,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 86, normalized size = 2.87

method	result	size
risch	\({\mathrm e}^{-\frac {\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x^{2}+\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x -\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )-x}{\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x-1)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-2*x-1)*exp(x)^2-2*x^2-x)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+
exp(x)^2+x)^2+(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*x*exp(2*
x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+x)/ln(exp
(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/ln(exp(2*x)^2+2*exp(x)*exp(2*
x)+exp(x)^2+x)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-2*x-1)*exp(x)^2-2*x^2-x)*log(exp(2*x)^2+2*exp(x)*ex
p(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*
x*exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+
x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*ex
p(x)*exp(2*x)+exp(x)^2+x)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

________________________________________________________________________________________

mupad [B]  time = 5.33, size = 35, normalized size = 1.17 \begin {gather*} {\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{\frac {x}{\ln \left (x+{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x - log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))*(x + x^2 - 1))/log(x + exp(2*x) + 2*exp(3*x) + exp(4*
x)))*(x + 2*x*exp(2*x) + 6*x*exp(3*x) + 4*x*exp(4*x) + log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))^2*(x + exp(2*
x)*(2*x + 1) + exp(4*x)*(2*x + 1) + exp(3*x)*(4*x + 2) + 2*x^2) - log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))*(x
+ exp(2*x) + 2*exp(3*x) + exp(4*x))))/(log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))^2*(x + exp(2*x) + 2*exp(3*x)
+ exp(4*x))),x)

[Out]

exp(-x)*exp(1)*exp(-x^2)*exp(x/log(x + exp(2*x) + 2*exp(3*x) + exp(4*x)))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x)**2+(-4*x-2)*exp(x)*exp(2*x)+(-2*x-1)*exp(x)**2-2*x**2-x)*ln(exp(2*x)**2+2*exp(x)
*exp(2*x)+exp(x)**2+x)**2+(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)*ln(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)*
*2+x)-4*x*exp(2*x)**2-6*x*exp(x)*exp(2*x)-2*x*exp(x)**2-x)*exp(((-x**2-x+1)*ln(exp(2*x)**2+2*exp(x)*exp(2*x)+e
xp(x)**2+x)+x)/ln(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x))/(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)/ln(e
xp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)**2,x)

[Out]

Timed out

________________________________________________________________________________________