3.100.61 \(\int \frac {-4+4 e^3-4 e^{e^x} x^4+e^{e^x} (40 x^3+10 e^x x^4) \log (1-e^3+e^{e^x} x^4)}{5-5 e^3+5 e^{e^x} x^4} \, dx\)

Optimal. Leaf size=26 \[ -4-\frac {4 x}{5}+\log ^2\left (1-e^3+e^{e^x} x^4\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 3.90, 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+4 e^3-4 e^{e^x} x^4+e^{e^x} \left (40 x^3+10 e^x x^4\right ) \log \left (1-e^3+e^{e^x} x^4\right )}{5-5 e^3+5 e^{e^x} x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + 4*E^3 - 4*E^E^x*x^4 + E^E^x*(40*x^3 + 10*E^x*x^4)*Log[1 - E^3 + E^E^x*x^4])/(5 - 5*E^3 + 5*E^E^x*x^4
),x]

[Out]

(-4*x)/5 + 8*(1 - E^3)*Log[1 - E^3 + E^E^x*x^4]*Defer[Int][1/(x*(-1 + E^3 - E^E^x*x^4)), x] + 2*Log[1 - E^3 +
E^E^x*x^4]*Defer[Int][(E^(E^x + x)*x^4)/(1 - E^3 + E^E^x*x^4), x] + 8*Defer[Int][Log[1 - E^3 + E^E^x*x^4]/x, x
] - 8*Defer[Int][(E^E^x*x^3*Defer[Int][(E^(E^x + x)*x^4)/(1 - E^3 + E^E^x*x^4), x])/(1 - E^3 + E^E^x*x^4), x]
- 2*Defer[Int][(E^(E^x + x)*x^4*Defer[Int][(E^(E^x + x)*x^4)/(1 - E^3 + E^E^x*x^4), x])/(1 - E^3 + E^E^x*x^4),
x] - 32*(1 - E^3)*Defer[Int][(E^E^x*x^3*Defer[Int][-((1 - E^3)*x + E^E^x*x^5)^(-1), x])/(1 - E^3 + E^E^x*x^4)
, x] - 8*(1 - E^3)*Defer[Int][(E^(E^x + x)*x^4*Defer[Int][-((1 - E^3)*x + E^E^x*x^5)^(-1), x])/(1 - E^3 + E^E^
x*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 \left (1-e^3\right )-4 e^{e^x} x^4+e^{e^x} \left (40 x^3+10 e^x x^4\right ) \log \left (1-e^3+e^{e^x} x^4\right )}{5 \left (1-e^3+e^{e^x} x^4\right )} \, dx\\ &=\frac {1}{5} \int \frac {-4 \left (1-e^3\right )-4 e^{e^x} x^4+e^{e^x} \left (40 x^3+10 e^x x^4\right ) \log \left (1-e^3+e^{e^x} x^4\right )}{1-e^3+e^{e^x} x^4} \, dx\\ &=\frac {1}{5} \int \left (\frac {10 e^{e^x+x} x^4 \log \left (1-e^3+e^{e^x} x^4\right )}{1-e^3+e^{e^x} x^4}-\frac {4 \left (-1+e^3-e^{e^x} x^4+10 e^{e^x} x^3 \log \left (1-e^3+e^{e^x} x^4\right )\right )}{-1+e^3-e^{e^x} x^4}\right ) \, dx\\ &=-\left (\frac {4}{5} \int \frac {-1+e^3-e^{e^x} x^4+10 e^{e^x} x^3 \log \left (1-e^3+e^{e^x} x^4\right )}{-1+e^3-e^{e^x} x^4} \, dx\right )+2 \int \frac {e^{e^x+x} x^4 \log \left (1-e^3+e^{e^x} x^4\right )}{1-e^3+e^{e^x} x^4} \, dx\\ &=-\left (\frac {4}{5} \int \left (\frac {x-10 \log \left (1-e^3+e^{e^x} x^4\right )}{x}+\frac {10 \left (-1+e^3\right ) \log \left (1-e^3+e^{e^x} x^4\right )}{x \left (-1+e^3-e^{e^x} x^4\right )}\right ) \, dx\right )-2 \int \frac {e^{e^x} x^3 \left (4+e^x x\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (2 \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx\\ &=-\left (\frac {4}{5} \int \frac {x-10 \log \left (1-e^3+e^{e^x} x^4\right )}{x} \, dx\right )-2 \int \left (\frac {4 e^{e^x} x^3 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4}+\frac {e^{e^x+x} x^4 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4}\right ) \, dx+\left (8 \left (1-e^3\right )\right ) \int \frac {\log \left (1-e^3+e^{e^x} x^4\right )}{x \left (-1+e^3-e^{e^x} x^4\right )} \, dx+\left (2 \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx\\ &=-\left (\frac {4}{5} \int \left (1-\frac {10 \log \left (1-e^3+e^{e^x} x^4\right )}{x}\right ) \, dx\right )-2 \int \frac {e^{e^x+x} x^4 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx-8 \int \frac {e^{e^x} x^3 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx-\left (8 \left (1-e^3\right )\right ) \int \frac {e^{e^x} x^3 \left (4+e^x x\right ) \int -\frac {1}{\left (1-e^3\right ) x+e^{e^x} x^5} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (2 \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx+\left (8 \left (1-e^3\right ) \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {1}{x \left (-1+e^3-e^{e^x} x^4\right )} \, dx\\ &=-\frac {4 x}{5}-2 \int \frac {e^{e^x+x} x^4 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx+8 \int \frac {\log \left (1-e^3+e^{e^x} x^4\right )}{x} \, dx-8 \int \frac {e^{e^x} x^3 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (8 \left (1-e^3\right )\right ) \int \left (\frac {4 e^{e^x} x^3 \int \frac {1}{\left (1-e^3\right ) x+e^{e^x} x^5} \, dx}{1-e^3+e^{e^x} x^4}+\frac {e^{e^x+x} x^4 \int \frac {1}{\left (1-e^3\right ) x+e^{e^x} x^5} \, dx}{1-e^3+e^{e^x} x^4}\right ) \, dx+\left (2 \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx+\left (8 \left (1-e^3\right ) \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {1}{x \left (-1+e^3-e^{e^x} x^4\right )} \, dx\\ &=-\frac {4 x}{5}-2 \int \frac {e^{e^x+x} x^4 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx+8 \int \frac {\log \left (1-e^3+e^{e^x} x^4\right )}{x} \, dx-8 \int \frac {e^{e^x} x^3 \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (8 \left (1-e^3\right )\right ) \int \frac {e^{e^x+x} x^4 \int \frac {1}{\left (1-e^3\right ) x+e^{e^x} x^5} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (32 \left (1-e^3\right )\right ) \int \frac {e^{e^x} x^3 \int \frac {1}{\left (1-e^3\right ) x+e^{e^x} x^5} \, dx}{1-e^3+e^{e^x} x^4} \, dx+\left (2 \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {e^{e^x+x} x^4}{1-e^3+e^{e^x} x^4} \, dx+\left (8 \left (1-e^3\right ) \log \left (1-e^3+e^{e^x} x^4\right )\right ) \int \frac {1}{x \left (-1+e^3-e^{e^x} x^4\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.38, size = 29, normalized size = 1.12 \begin {gather*} \frac {1}{5} \left (-4 x+5 \log ^2\left (1-e^3+e^{e^x} x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*E^3 - 4*E^E^x*x^4 + E^E^x*(40*x^3 + 10*E^x*x^4)*Log[1 - E^3 + E^E^x*x^4])/(5 - 5*E^3 + 5*E^E
^x*x^4),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 20, normalized size = 0.77 \begin {gather*} \log \left (x^{4} e^{\left (e^{x}\right )} - e^{3} + 1\right )^{2} - \frac {4}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*exp(x)*x^4+40*x^3)*exp(exp(x))*log(x^4*exp(exp(x))-exp(3)+1)-4*x^4*exp(exp(x))+4*exp(3)-4)/(5*x
^4*exp(exp(x))-5*exp(3)+5),x, algorithm="fricas")

[Out]

log(x^4*e^(e^x) - e^3 + 1)^2 - 4/5*x

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (2 \, x^{4} e^{\left (e^{x}\right )} - 5 \, {\left (x^{4} e^{x} + 4 \, x^{3}\right )} e^{\left (e^{x}\right )} \log \left (x^{4} e^{\left (e^{x}\right )} - e^{3} + 1\right ) - 2 \, e^{3} + 2\right )}}{5 \, {\left (x^{4} e^{\left (e^{x}\right )} - e^{3} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*exp(x)*x^4+40*x^3)*exp(exp(x))*log(x^4*exp(exp(x))-exp(3)+1)-4*x^4*exp(exp(x))+4*exp(3)-4)/(5*x
^4*exp(exp(x))-5*exp(3)+5),x, algorithm="giac")

[Out]

integrate(-2/5*(2*x^4*e^(e^x) - 5*(x^4*e^x + 4*x^3)*e^(e^x)*log(x^4*e^(e^x) - e^3 + 1) - 2*e^3 + 2)/(x^4*e^(e^
x) - e^3 + 1), x)

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (10 \,{\mathrm e}^{x} x^{4}+40 x^{3}\right ) {\mathrm e}^{{\mathrm e}^{x}} \ln \left (x^{4} {\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{3}+1\right )-4 x^{4} {\mathrm e}^{{\mathrm e}^{x}}+4 \,{\mathrm e}^{3}-4}{5 x^{4} {\mathrm e}^{{\mathrm e}^{x}}-5 \,{\mathrm e}^{3}+5}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*exp(x)*x^4+40*x^3)*exp(exp(x))*ln(x^4*exp(exp(x))-exp(3)+1)-4*x^4*exp(exp(x))+4*exp(3)-4)/(5*x^4*exp(
exp(x))-5*exp(3)+5),x)

[Out]

int(((10*exp(x)*x^4+40*x^3)*exp(exp(x))*ln(x^4*exp(exp(x))-exp(3)+1)-4*x^4*exp(exp(x))+4*exp(3)-4)/(5*x^4*exp(
exp(x))-5*exp(3)+5),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2}{5} \, \int \frac {2 \, x^{4} e^{\left (e^{x}\right )} - 5 \, {\left (x^{4} e^{x} + 4 \, x^{3}\right )} e^{\left (e^{x}\right )} \log \left (x^{4} e^{\left (e^{x}\right )} - e^{3} + 1\right ) - 2 \, e^{3} + 2}{x^{4} e^{\left (e^{x}\right )} - e^{3} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*exp(x)*x^4+40*x^3)*exp(exp(x))*log(x^4*exp(exp(x))-exp(3)+1)-4*x^4*exp(exp(x))+4*exp(3)-4)/(5*x
^4*exp(exp(x))-5*exp(3)+5),x, algorithm="maxima")

[Out]

-2/5*integrate((2*x^4*e^(e^x) - 5*(x^4*e^x + 4*x^3)*e^(e^x)*log(x^4*e^(e^x) - e^3 + 1) - 2*e^3 + 2)/(x^4*e^(e^
x) - e^3 + 1), x)

________________________________________________________________________________________

mupad [B]  time = 0.46, size = 20, normalized size = 0.77 \begin {gather*} {\ln \left (x^4\,{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^3+1\right )}^2-\frac {4\,x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(3) - 4*x^4*exp(exp(x)) + log(x^4*exp(exp(x)) - exp(3) + 1)*exp(exp(x))*(10*x^4*exp(x) + 40*x^3) - 4
)/(5*x^4*exp(exp(x)) - 5*exp(3) + 5),x)

[Out]

log(x^4*exp(exp(x)) - exp(3) + 1)^2 - (4*x)/5

________________________________________________________________________________________

sympy [A]  time = 0.70, size = 20, normalized size = 0.77 \begin {gather*} - \frac {4 x}{5} + \log {\left (x^{4} e^{e^{x}} - e^{3} + 1 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*exp(x)*x**4+40*x**3)*exp(exp(x))*ln(x**4*exp(exp(x))-exp(3)+1)-4*x**4*exp(exp(x))+4*exp(3)-4)/(
5*x**4*exp(exp(x))-5*exp(3)+5),x)

[Out]

-4*x/5 + log(x**4*exp(exp(x)) - exp(3) + 1)**2

________________________________________________________________________________________