3.14.95 \(\int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+(20+4 x^2-16 x^3+12 x^4) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+(e^2 (25-5 x)+5 e^2 \log (x)) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {-5+\left (x-x^2\right )^2-\frac {5}{4} e^2 \log ^2(5-x+\log (x))}{x} \]

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Rubi [F]  time = 2.04, 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 {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 + 10*x)
*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Log[x]),
x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{4 x^2 (5-x+\log (x))} \, dx\\ &=\frac {1}{4} \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{x^2 (5-x+\log (x))} \, dx\\ &=\frac {1}{4} \int \left (-\frac {20}{-5+x-\log (x)}-\frac {100}{x^2 (-5+x-\log (x))}+\frac {20}{x (-5+x-\log (x))}+\frac {84 x}{-5+x-\log (x)}-\frac {76 x^2}{-5+x-\log (x)}+\frac {12 x^3}{-5+x-\log (x)}-\frac {4 \left (5+x^2-4 x^3+3 x^4\right ) \log (x)}{x^2 (-5+x-\log (x))}-\frac {10 e^2 (-1+x) \log (5-x+\log (x))}{x^2 (-5+x-\log (x))}+\frac {5 e^2 \log ^2(5-x+\log (x))}{x^2}\right ) \, dx\\ &=3 \int \frac {x^3}{-5+x-\log (x)} \, dx-5 \int \frac {1}{-5+x-\log (x)} \, dx+5 \int \frac {1}{x (-5+x-\log (x))} \, dx-19 \int \frac {x^2}{-5+x-\log (x)} \, dx+21 \int \frac {x}{-5+x-\log (x)} \, dx-25 \int \frac {1}{x^2 (-5+x-\log (x))} \, dx+\frac {1}{4} \left (5 e^2\right ) \int \frac {\log ^2(5-x+\log (x))}{x^2} \, dx-\frac {1}{2} \left (5 e^2\right ) \int \frac {(-1+x) \log (5-x+\log (x))}{x^2 (-5+x-\log (x))} \, dx-\int \frac {\left (5+x^2-4 x^3+3 x^4\right ) \log (x)}{x^2 (-5+x-\log (x))} \, dx\\ &=3 \int \frac {x^3}{-5+x-\log (x)} \, dx-5 \int \frac {1}{-5+x-\log (x)} \, dx+5 \int \frac {1}{x (-5+x-\log (x))} \, dx-19 \int \frac {x^2}{-5+x-\log (x)} \, dx+21 \int \frac {x}{-5+x-\log (x)} \, dx-25 \int \frac {1}{x^2 (-5+x-\log (x))} \, dx+\frac {1}{4} \left (5 e^2\right ) \int \frac {\log ^2(5-x+\log (x))}{x^2} \, dx-\frac {1}{2} \left (5 e^2\right ) \int \left (-\frac {\log (5-x+\log (x))}{x^2 (-5+x-\log (x))}+\frac {\log (5-x+\log (x))}{x (-5+x-\log (x))}\right ) \, dx-\int \left (\frac {-5-x^2+4 x^3-3 x^4}{x^2}+\frac {-25+5 x-5 x^2+21 x^3-19 x^4+3 x^5}{x^2 (-5+x-\log (x))}\right ) \, dx\\ &=3 \int \frac {x^3}{-5+x-\log (x)} \, dx-5 \int \frac {1}{-5+x-\log (x)} \, dx+5 \int \frac {1}{x (-5+x-\log (x))} \, dx-19 \int \frac {x^2}{-5+x-\log (x)} \, dx+21 \int \frac {x}{-5+x-\log (x)} \, dx-25 \int \frac {1}{x^2 (-5+x-\log (x))} \, dx+\frac {1}{4} \left (5 e^2\right ) \int \frac {\log ^2(5-x+\log (x))}{x^2} \, dx+\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x^2 (-5+x-\log (x))} \, dx-\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x (-5+x-\log (x))} \, dx-\int \frac {-5-x^2+4 x^3-3 x^4}{x^2} \, dx-\int \frac {-25+5 x-5 x^2+21 x^3-19 x^4+3 x^5}{x^2 (-5+x-\log (x))} \, dx\\ &=3 \int \frac {x^3}{-5+x-\log (x)} \, dx-5 \int \frac {1}{-5+x-\log (x)} \, dx+5 \int \frac {1}{x (-5+x-\log (x))} \, dx-19 \int \frac {x^2}{-5+x-\log (x)} \, dx+21 \int \frac {x}{-5+x-\log (x)} \, dx-25 \int \frac {1}{x^2 (-5+x-\log (x))} \, dx+\frac {1}{4} \left (5 e^2\right ) \int \frac {\log ^2(5-x+\log (x))}{x^2} \, dx+\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x^2 (-5+x-\log (x))} \, dx-\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x (-5+x-\log (x))} \, dx-\int \left (-1-\frac {5}{x^2}+4 x-3 x^2\right ) \, dx-\int \left (-\frac {5}{-5+x-\log (x)}-\frac {25}{x^2 (-5+x-\log (x))}+\frac {5}{x (-5+x-\log (x))}+\frac {21 x}{-5+x-\log (x)}-\frac {19 x^2}{-5+x-\log (x)}+\frac {3 x^3}{-5+x-\log (x)}\right ) \, dx\\ &=-\frac {5}{x}+x-2 x^2+x^3+\frac {1}{4} \left (5 e^2\right ) \int \frac {\log ^2(5-x+\log (x))}{x^2} \, dx+\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x^2 (-5+x-\log (x))} \, dx-\frac {1}{2} \left (5 e^2\right ) \int \frac {\log (5-x+\log (x))}{x (-5+x-\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 +
10*x)*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Lo
g[x]),x]

[Out]

Integrate[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 +
10*x)*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Lo
g[x]), x]

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fricas [A]  time = 0.96, size = 36, normalized size = 1.12 \begin {gather*} \frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \relax (x) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)*exp(2)*log(log(x)-x+5)+(12*x^4-16*x^
3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 36, normalized size = 1.12 \begin {gather*} \frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \relax (x) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)*exp(2)*log(log(x)-x+5)+(12*x^4-16*x^
3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 36, normalized size = 1.12

method	result	size
risch	\(-\frac {5 \,{\mathrm e}^{2} \ln \left (\ln \relax (x )-x +5\right )^{2}}{4 x}+\frac {x^{4}-2 x^{3}+x^{2}-5}{x}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*exp(2)*ln(x)+(-5*x+25)*exp(2))*ln(ln(x)-x+5)^2+(10*x-10)*exp(2)*ln(ln(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)
*ln(x)-12*x^5+76*x^4-84*x^3+20*x^2-20*x+100)/(4*x^2*ln(x)-4*x^3+20*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.70, size = 36, normalized size = 1.12 \begin {gather*} \frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \relax (x) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)*exp(2)*log(log(x)-x+5)+(12*x^4-16*x^
3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.12, size = 32, normalized size = 1.00 \begin {gather*} x-\frac {5}{x}-2\,x^2+x^3-\frac {5\,{\mathrm {e}}^2\,{\ln \left (\ln \relax (x)-x+5\right )}^2}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(4*x^2 - 16*x^3 + 12*x^4 + 20) - 20*x + log(log(x) - x + 5)^2*(5*exp(2)*log(x) - exp(2)*(5*x - 25)
) + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + exp(2)*log(log(x) - x + 5)*(10*x - 10) + 100)/(4*x^2*log(x) + 20*x^2 -
4*x^3),x)

[Out]

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

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sympy [A]  time = 0.46, size = 31, normalized size = 0.97 \begin {gather*} x^{3} - 2 x^{2} + x - \frac {5 e^{2} \log {\left (- x + \log {\relax (x )} + 5 \right )}^{2}}{4 x} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(2)*ln(x)+(-5*x+25)*exp(2))*ln(ln(x)-x+5)**2+(10*x-10)*exp(2)*ln(ln(x)-x+5)+(12*x**4-16*x**3+
4*x**2+20)*ln(x)-12*x**5+76*x**4-84*x**3+20*x**2-20*x+100)/(4*x**2*ln(x)-4*x**3+20*x**2),x)

[Out]

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

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