3.72.18 \(\int \frac {4+\log (3)+e^{e^x+x+x^4} (-6+x-20 x^3+4 x^4+e^x (-5+x-\log (3))+(-1-4 x^3) \log (3))}{1+e^{2 e^x+2 x+2 x^4}+e^{e^x+x+x^4} (2-2 x)-2 x+x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac {5-x+\log (3)}{1+e^{e^x+x+x^4}-x} \]

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Rubi [F]  time = 4.11, 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 {4+\log (3)+e^{e^x+x+x^4} \left (-6+x-20 x^3+4 x^4+e^x (-5+x-\log (3))+\left (-1-4 x^3\right ) \log (3)\right )}{1+e^{2 e^x+2 x+2 x^4}+e^{e^x+x+x^4} (2-2 x)-2 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + Log[3] + E^(E^x + x + x^4)*(-6 + x - 20*x^3 + 4*x^4 + E^x*(-5 + x - Log[3]) + (-1 - 4*x^3)*Log[3]))/(
1 + E^(2*E^x + 2*x + 2*x^4) + E^(E^x + x + x^4)*(2 - 2*x) - 2*x + x^2),x]

[Out]

2*(5 + Log[3])*Defer[Int][(1 + E^(E^x + x + x^4) - x)^(-2), x] + (5 + Log[3])*Defer[Int][E^x/(1 + E^(E^x + x +
x^4) - x)^2, x] - (6 + Log[3])*Defer[Int][(1 + E^(E^x + x + x^4) - x)^(-1), x] - (5 + Log[3])*Defer[Int][E^x/
(1 + E^(E^x + x + x^4) - x), x] - 2*Defer[Int][x/(1 + E^(E^x + x + x^4) - x)^2, x] - (5 + Log[3])*Defer[Int][x
/(1 + E^(E^x + x + x^4) - x)^2, x] - Defer[Int][(E^x*x)/(1 + E^(E^x + x + x^4) - x)^2, x] - (5 + Log[3])*Defer
[Int][(E^x*x)/(1 + E^(E^x + x + x^4) - x)^2, x] + Defer[Int][x/(1 + E^(E^x + x + x^4) - x), x] + Defer[Int][(E
^x*x)/(1 + E^(E^x + x + x^4) - x), x] + Defer[Int][x^2/(1 + E^(E^x + x + x^4) - x)^2, x] + Defer[Int][(E^x*x^2
)/(1 + E^(E^x + x + x^4) - x)^2, x] + 4*(5 + Log[3])*Defer[Int][x^3/(1 + E^(E^x + x + x^4) - x)^2, x] - 4*(5 +
Log[3])*Defer[Int][x^3/(1 + E^(E^x + x + x^4) - x), x] - 4*Defer[Int][x^4/(1 + E^(E^x + x + x^4) - x)^2, x] -
4*(5 + Log[3])*Defer[Int][x^4/(1 + E^(E^x + x + x^4) - x)^2, x] + 4*Defer[Int][x^4/(1 + E^(E^x + x + x^4) - x
), x] + 4*Defer[Int][x^5/(1 + E^(E^x + x + x^4) - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (1+\frac {\log (3)}{4}\right )+e^{e^x+x+x^4} \left (-6+x-20 x^3+4 x^4+e^x (-5+x-\log (3))+\left (-1-4 x^3\right ) \log (3)\right )}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx\\ &=\int \left (\frac {x+e^x x+4 x^4-6 \left (1+\frac {\log (3)}{6}\right )-5 e^x \left (1+\frac {\log (3)}{5}\right )-20 x^3 \left (1+\frac {\log (3)}{5}\right )}{1+e^{e^x+x+x^4}-x}+\frac {\left (-2-e^x+x+e^x x-4 x^3+4 x^4\right ) (-5+x-\log (3))}{\left (1+e^{e^x+x+x^4}-x\right )^2}\right ) \, dx\\ &=\int \frac {x+e^x x+4 x^4-6 \left (1+\frac {\log (3)}{6}\right )-5 e^x \left (1+\frac {\log (3)}{5}\right )-20 x^3 \left (1+\frac {\log (3)}{5}\right )}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {\left (-2-e^x+x+e^x x-4 x^3+4 x^4\right ) (-5+x-\log (3))}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx\\ &=\int \left (\frac {x \left (-2-e^x+x+e^x x-4 x^3+4 x^4\right )}{\left (1+e^{e^x+x+x^4}-x\right )^2}-\frac {5 \left (-2-e^x+x+e^x x-4 x^3+4 x^4\right ) \left (1+\frac {\log (3)}{5}\right )}{\left (1+e^{e^x+x+x^4}-x\right )^2}\right ) \, dx+\int \left (\frac {x}{1+e^{e^x+x+x^4}-x}+\frac {e^x x}{1+e^{e^x+x+x^4}-x}+\frac {4 x^4}{1+e^{e^x+x+x^4}-x}-\frac {e^x (5+\log (3))}{1+e^{e^x+x+x^4}-x}-\frac {4 x^3 (5+\log (3))}{1+e^{e^x+x+x^4}-x}-\frac {6+\log (3)}{1+e^{e^x+x+x^4}-x}\right ) \, dx\\ &=4 \int \frac {x^4}{1+e^{e^x+x+x^4}-x} \, dx+(-6-\log (3)) \int \frac {1}{1+e^{e^x+x+x^4}-x} \, dx+(-5-\log (3)) \int \frac {e^x}{1+e^{e^x+x+x^4}-x} \, dx-(5+\log (3)) \int \frac {-2-e^x+x+e^x x-4 x^3+4 x^4}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx-(4 (5+\log (3))) \int \frac {x^3}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {x}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {e^x x}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {x \left (-2-e^x+x+e^x x-4 x^3+4 x^4\right )}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx\\ &=4 \int \frac {x^4}{1+e^{e^x+x+x^4}-x} \, dx+(-6-\log (3)) \int \frac {1}{1+e^{e^x+x+x^4}-x} \, dx+(-5-\log (3)) \int \frac {e^x}{1+e^{e^x+x+x^4}-x} \, dx-(5+\log (3)) \int \left (-\frac {2}{\left (1+e^{e^x+x+x^4}-x\right )^2}-\frac {e^x}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {x}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {e^x x}{\left (1+e^{e^x+x+x^4}-x\right )^2}-\frac {4 x^3}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {4 x^4}{\left (1+e^{e^x+x+x^4}-x\right )^2}\right ) \, dx-(4 (5+\log (3))) \int \frac {x^3}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {x}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {e^x x}{1+e^{e^x+x+x^4}-x} \, dx+\int \left (-\frac {2 x}{\left (1+e^{e^x+x+x^4}-x\right )^2}-\frac {e^x x}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {x^2}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {e^x x^2}{\left (1+e^{e^x+x+x^4}-x\right )^2}-\frac {4 x^4}{\left (1+e^{e^x+x+x^4}-x\right )^2}+\frac {4 x^5}{\left (1+e^{e^x+x+x^4}-x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx\right )-4 \int \frac {x^4}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+4 \int \frac {x^4}{1+e^{e^x+x+x^4}-x} \, dx+4 \int \frac {x^5}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+(-6-\log (3)) \int \frac {1}{1+e^{e^x+x+x^4}-x} \, dx-(-5-\log (3)) \int \frac {e^x}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+(-5-\log (3)) \int \frac {e^x}{1+e^{e^x+x+x^4}-x} \, dx-(5+\log (3)) \int \frac {x}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx-(5+\log (3)) \int \frac {e^x x}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+(2 (5+\log (3))) \int \frac {1}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+(4 (5+\log (3))) \int \frac {x^3}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx-(4 (5+\log (3))) \int \frac {x^3}{1+e^{e^x+x+x^4}-x} \, dx-(4 (5+\log (3))) \int \frac {x^4}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx-\int \frac {e^x x}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+\int \frac {x}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {e^x x}{1+e^{e^x+x+x^4}-x} \, dx+\int \frac {x^2}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx+\int \frac {e^x x^2}{\left (1+e^{e^x+x+x^4}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.18, size = 25, normalized size = 1.00 \begin {gather*} \frac {5-x+\log (3)}{1+e^{e^x+x+x^4}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + Log[3] + E^(E^x + x + x^4)*(-6 + x - 20*x^3 + 4*x^4 + E^x*(-5 + x - Log[3]) + (-1 - 4*x^3)*Log[
3]))/(1 + E^(2*E^x + 2*x + 2*x^4) + E^(E^x + x + x^4)*(2 - 2*x) - 2*x + x^2),x]

[Out]

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

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fricas [A]  time = 0.73, size = 23, normalized size = 0.92 \begin {gather*} \frac {x - \log \relax (3) - 5}{x - e^{\left (x^{4} + x + e^{x}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(3)+x-5)*exp(x)+(-4*x^3-1)*log(3)+4*x^4-20*x^3+x-6)*exp(exp(x)+x^4+x)+4+log(3))/(exp(exp(x)+x
^4+x)^2+(-2*x+2)*exp(exp(x)+x^4+x)+x^2-2*x+1),x, algorithm="fricas")

[Out]

(x - log(3) - 5)/(x - e^(x^4 + x + e^x) - 1)

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giac [A]  time = 0.24, size = 23, normalized size = 0.92 \begin {gather*} \frac {x - \log \relax (3) - 5}{x - e^{\left (x^{4} + x + e^{x}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(3)+x-5)*exp(x)+(-4*x^3-1)*log(3)+4*x^4-20*x^3+x-6)*exp(exp(x)+x^4+x)+4+log(3))/(exp(exp(x)+x
^4+x)^2+(-2*x+2)*exp(exp(x)+x^4+x)+x^2-2*x+1),x, algorithm="giac")

[Out]

(x - log(3) - 5)/(x - e^(x^4 + x + e^x) - 1)

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maple [A]  time = 0.16, size = 25, normalized size = 1.00

method	result	size
risch	\(-\frac {\ln \relax (3)-x +5}{x -{\mathrm e}^{{\mathrm e}^{x}+x^{4}+x}-1}\)	\(25\)
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{x}+x^{4}+x}-4-\ln \relax (3)}{x -{\mathrm e}^{{\mathrm e}^{x}+x^{4}+x}-1}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-ln(3)+x-5)*exp(x)+(-4*x^3-1)*ln(3)+4*x^4-20*x^3+x-6)*exp(exp(x)+x^4+x)+4+ln(3))/(exp(exp(x)+x^4+x)^2+(
-2*x+2)*exp(exp(x)+x^4+x)+x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

-(ln(3)-x+5)/(x-exp(exp(x)+x^4+x)-1)

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maxima [A]  time = 0.50, size = 23, normalized size = 0.92 \begin {gather*} \frac {x - \log \relax (3) - 5}{x - e^{\left (x^{4} + x + e^{x}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(3)+x-5)*exp(x)+(-4*x^3-1)*log(3)+4*x^4-20*x^3+x-6)*exp(exp(x)+x^4+x)+4+log(3))/(exp(exp(x)+x
^4+x)^2+(-2*x+2)*exp(exp(x)+x^4+x)+x^2-2*x+1),x, algorithm="maxima")

[Out]

(x - log(3) - 5)/(x - e^(x^4 + x + e^x) - 1)

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mupad [B]  time = 4.50, size = 113, normalized size = 4.52 \begin {gather*} -\frac {2\,\ln \relax (3)-7\,x+5\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^x-x\,\ln \relax (3)+4\,x^3\,\ln \relax (3)-4\,x^4\,\ln \relax (3)+{\mathrm {e}}^x\,\ln \relax (3)-6\,x\,{\mathrm {e}}^x+x^2+20\,x^3-24\,x^4+4\,x^5-x\,{\mathrm {e}}^x\,\ln \relax (3)+10}{\left ({\mathrm {e}}^{x+{\mathrm {e}}^x+x^4}-x+1\right )\,\left (x-{\mathrm {e}}^x+x\,{\mathrm {e}}^x-4\,x^3+4\,x^4-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3) - exp(x + exp(x) + x^4)*(log(3)*(4*x^3 + 1) - x + 20*x^3 - 4*x^4 + exp(x)*(log(3) - x + 5) + 6) +
4)/(exp(2*x + 2*exp(x) + 2*x^4) - 2*x - exp(x + exp(x) + x^4)*(2*x - 2) + x^2 + 1),x)

[Out]

-(2*log(3) - 7*x + 5*exp(x) + x^2*exp(x) - x*log(3) + 4*x^3*log(3) - 4*x^4*log(3) + exp(x)*log(3) - 6*x*exp(x)
+ x^2 + 20*x^3 - 24*x^4 + 4*x^5 - x*exp(x)*log(3) + 10)/((exp(x + exp(x) + x^4) - x + 1)*(x - exp(x) + x*exp(
x) - 4*x^3 + 4*x^4 - 2))

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sympy [A]  time = 0.22, size = 19, normalized size = 0.76 \begin {gather*} \frac {- x + \log {\relax (3 )} + 5}{- x + e^{x^{4} + x + e^{x}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-ln(3)+x-5)*exp(x)+(-4*x**3-1)*ln(3)+4*x**4-20*x**3+x-6)*exp(exp(x)+x**4+x)+4+ln(3))/(exp(exp(x)+
x**4+x)**2+(-2*x+2)*exp(exp(x)+x**4+x)+x**2-2*x+1),x)

[Out]

(-x + log(3) + 5)/(-x + exp(x**4 + x + exp(x)) + 1)

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