3.96.51 \(\int \frac {20 x^2-20 x^3+5 x^4+e (-20+20 x-5 x^2)+e^{2 e^4-2 x+(-e^4+x) \log ^4(x)} (3 x^2-2 x^3+(-8 x^2+4 x^3+e^4 (8 x-4 x^2)) \log ^3(x)+(-2 x^2+x^3) \log ^4(x))}{20 x^2-20 x^3+5 x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac {e^{\left (-e^4+x\right ) \left (-2+\log ^4(x)\right )}}{5 (2-x)}+\frac {e}{x}+x \]

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Rubi [F]  time = 6.19, 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 {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*x^2 - 20*x^3 + 5*x^4 + E*(-20 + 20*x - 5*x^2) + E^(2*E^4 - 2*x + (-E^4 + x)*Log[x]^4)*(3*x^2 - 2*x^3 +
(-8*x^2 + 4*x^3 + E^4*(8*x - 4*x^2))*Log[x]^3 + (-2*x^2 + x^3)*Log[x]^4))/(20*x^2 - 20*x^3 + 5*x^4),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{x^2 \left (20-20 x+5 x^2\right )} \, dx\\ &=\int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{5 (-2+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{(-2+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {5 \left (e-x^2\right )}{x^2}+\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left (3 x-2 x^2+8 e^4 \log ^3(x)-8 \left (1+\frac {e^4}{2}\right ) x \log ^3(x)+4 x^2 \log ^3(x)-2 x \log ^4(x)+x^2 \log ^4(x)\right )}{(2-x)^2 x}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left (3 x-2 x^2+8 e^4 \log ^3(x)-8 \left (1+\frac {e^4}{2}\right ) x \log ^3(x)+4 x^2 \log ^3(x)-2 x \log ^4(x)+x^2 \log ^4(x)\right )}{(2-x)^2 x} \, dx-\int \frac {e-x^2}{x^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left ((3-2 x) x+4 (-2+x) \left (-e^4+x\right ) \log ^3(x)+(-2+x) x \log ^4(x)\right )}{(2-x)^2 x} \, dx-\int \left (-1+\frac {e}{x^2}\right ) \, dx\\ &=\frac {e}{x}+x+\frac {1}{5} \int \left (\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} (3-2 x)}{(-2+x)^2}+\frac {4 e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left (-e^4+x\right ) \log ^3(x)}{(-2+x) x}+\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \log ^4(x)}{-2+x}\right ) \, dx\\ &=\frac {e}{x}+x+\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} (3-2 x)}{(-2+x)^2} \, dx+\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \log ^4(x)}{-2+x} \, dx+\frac {4}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left (-e^4+x\right ) \log ^3(x)}{(-2+x) x} \, dx\\ &=\frac {e}{x}+x+\frac {1}{5} \int \left (-\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{(-2+x)^2}-\frac {2 e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{-2+x}\right ) \, dx+\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \log ^4(x)}{-2+x} \, dx+\frac {4}{5} \int \left (-\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \left (-2+e^4\right ) \log ^3(x)}{2 (-2+x)}+\frac {e^{4-\left (e^4-x\right ) \left (-2+\log ^4(x)\right )} \log ^3(x)}{2 x}\right ) \, dx\\ &=\frac {e}{x}+x-\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{(-2+x)^2} \, dx+\frac {1}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \log ^4(x)}{-2+x} \, dx-\frac {2}{5} \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{-2+x} \, dx+\frac {2}{5} \int \frac {e^{4-\left (e^4-x\right ) \left (-2+\log ^4(x)\right )} \log ^3(x)}{x} \, dx+\frac {1}{5} \left (2 \left (2-e^4\right )\right ) \int \frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )} \log ^3(x)}{-2+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 37, normalized size = 1.09 \begin {gather*} \frac {1}{5} \left (\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{-2+x}+\frac {5 e}{x}+5 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*x^2 - 20*x^3 + 5*x^4 + E*(-20 + 20*x - 5*x^2) + E^(2*E^4 - 2*x + (-E^4 + x)*Log[x]^4)*(3*x^2 - 2
*x^3 + (-8*x^2 + 4*x^3 + E^4*(8*x - 4*x^2))*Log[x]^3 + (-2*x^2 + x^3)*Log[x]^4))/(20*x^2 - 20*x^3 + 5*x^4),x]

[Out]

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

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fricas [A]  time = 0.62, size = 51, normalized size = 1.50 \begin {gather*} \frac {5 \, x^{3} - 10 \, x^{2} + 5 \, {\left (x - 2\right )} e + x e^{\left ({\left (x - e^{4}\right )} \log \relax (x)^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x^{2} - 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)
^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="fricas")

[Out]

1/5*(5*x^3 - 10*x^2 + 5*(x - 2)*e + x*e^((x - e^4)*log(x)^4 - 2*x + 2*e^4))/(x^2 - 2*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, x^{4} - 20 \, x^{3} + 20 \, x^{2} - 5 \, {\left (x^{2} - 4 \, x + 4\right )} e + {\left ({\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)^{4} + 4 \, {\left (x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x\right )} e^{4}\right )} \log \relax (x)^{3} - 2 \, x^{3} + 3 \, x^{2}\right )} e^{\left ({\left (x - e^{4}\right )} \log \relax (x)^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)
^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="giac")

[Out]

integrate(1/5*(5*x^4 - 20*x^3 + 20*x^2 - 5*(x^2 - 4*x + 4)*e + ((x^3 - 2*x^2)*log(x)^4 + 4*(x^3 - 2*x^2 - (x^2
- 2*x)*e^4)*log(x)^3 - 2*x^3 + 3*x^2)*e^((x - e^4)*log(x)^4 - 2*x + 2*e^4))/(x^4 - 4*x^3 + 4*x^2), x)

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maple [A]  time = 0.19, size = 31, normalized size = 0.91

method	result	size
risch	\(x +\frac {{\mathrm e}}{x}+\frac {{\mathrm e}^{-\left (\ln \relax (x )^{4}-2\right ) \left ({\mathrm e}^{4}-x \right )}}{5 x -10}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-2*x^2)*ln(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*ln(x)^3-2*x^3+3*x^2)*exp((x-exp(4))*ln(x)^4+2*exp(
4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.55, size = 86, normalized size = 2.53 \begin {gather*} {\left (\frac {2 \, {\left (x - 1\right )}}{x^{2} - 2 \, x} + \log \left (x - 2\right ) - \log \relax (x)\right )} e - {\left (\frac {2}{x - 2} + \log \left (x - 2\right ) - \log \relax (x)\right )} e + x + \frac {e}{x - 2} + \frac {e^{\left (x \log \relax (x)^{4} - e^{4} \log \relax (x)^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)
^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 9.73, size = 41, normalized size = 1.21 \begin {gather*} x+\frac {\mathrm {e}}{x}+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-{\mathrm {e}}^4\,{\ln \relax (x)}^4}\,{\mathrm {e}}^{x\,{\ln \relax (x)}^4}}{5\,\left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1)*(5*x^2 - 20*x + 20) - 20*x^2 + 20*x^3 - 5*x^4 + exp(2*exp(4) - 2*x + log(x)^4*(x - exp(4)))*(log(
x)^4*(2*x^2 - x^3) - log(x)^3*(exp(4)*(8*x - 4*x^2) - 8*x^2 + 4*x^3) - 3*x^2 + 2*x^3))/(20*x^2 - 20*x^3 + 5*x^
4),x)

[Out]

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

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sympy [A]  time = 0.40, size = 31, normalized size = 0.91 \begin {gather*} x + \frac {e^{- 2 x + \left (x - e^{4}\right ) \log {\relax (x )}^{4} + 2 e^{4}}}{5 x - 10} + \frac {e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-2*x**2)*ln(x)**4+((-4*x**2+8*x)*exp(4)+4*x**3-8*x**2)*ln(x)**3-2*x**3+3*x**2)*exp((x-exp(4))
*ln(x)**4+2*exp(4)-2*x)+(-5*x**2+20*x-20)*exp(1)+5*x**4-20*x**3+20*x**2)/(5*x**4-20*x**3+20*x**2),x)

[Out]

x + exp(-2*x + (x - exp(4))*log(x)**4 + 2*exp(4))/(5*x - 10) + E/x

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