3.60.56 \(\int \frac {(8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 (2 x+16 x^2+16 x^3)) \log (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3})+(-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} (-8 x+2 x^2)+e^5 (-2 x-16 x^2-12 x^3+4 x^4)) \log ^2(\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3})}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 (-1-8 x-6 x^2+2 x^3)} \, dx\)

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

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[((8*E^10*x + 4*x^2 + 14*x^3 + 16*x^4 + 8*x^5 + E^5*(2*x + 16*x^2 + 16*x^3))*Log[(1 + E^5*(4 - x) + 4*x + 3
*x^2 - x^3)/(E^5*x + x^2 + x^3)] + (-2*x^2 - 10*x^3 - 14*x^4 - 4*x^5 + 2*x^6 + E^10*(-8*x + 2*x^2) + E^5*(-2*x
- 16*x^2 - 12*x^3 + 4*x^4))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)]^2)/(E^10*(-4 + x)
- x - 5*x^2 - 7*x^3 - 2*x^4 + x^5 + E^5*(-1 - 8*x - 6*x^2 + 2*x^3)),x]

[Out]

$Aborted

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}+2 x \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )\right ) \, dx\\ &=2 \int \frac {x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx+2 \int x \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx\\ &=x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-2 \int \frac {x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx+2 \int \left (4 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+\frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2}+\frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx\\ &=x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+2 \int \frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2} \, dx+2 \int \frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-2 \int \left (4 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+\frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2}+\frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx+8 \int \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx\\ &=8 x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-2 \int \frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2} \, dx-2 \int \frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+2 \int \left (\frac {\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1-i \sqrt {-1+4 e^5}+2 x}+\frac {\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+i \sqrt {-1+4 e^5}+2 x}\right ) \, dx+2 \int \left (\frac {3 \left (-1-4 e^5\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {3 \left (-5-3 e^5\right ) x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {\left (-17+2 e^5\right ) x^2 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx-8 \int \frac {-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx-8 \int \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.17, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((8*E^10*x + 4*x^2 + 14*x^3 + 16*x^4 + 8*x^5 + E^5*(2*x + 16*x^2 + 16*x^3))*Log[(1 + E^5*(4 - x) + 4
*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)] + (-2*x^2 - 10*x^3 - 14*x^4 - 4*x^5 + 2*x^6 + E^10*(-8*x + 2*x^2) + E^5
*(-2*x - 16*x^2 - 12*x^3 + 4*x^4))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)]^2)/(E^10*(-4
+ x) - x - 5*x^2 - 7*x^3 - 2*x^4 + x^5 + E^5*(-1 - 8*x - 6*x^2 + 2*x^3)),x]

[Out]

$Aborted

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fricas [A]  time = 0.80, size = 41, normalized size = 1.58 \begin {gather*} x^{2} \log \left (-\frac {x^{3} - 3 \, x^{2} + {\left (x - 4\right )} e^{5} - 4 \, x - 1}{x^{3} + x^{2} + x e^{5}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 4.48, size = 43, normalized size = 1.65 \begin {gather*} x^{2} \log \left (-\frac {x^{3} - 3 \, x^{2} + x e^{5} - 4 \, x - 4 \, e^{5} - 1}{x^{3} + x^{2} + x e^{5}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.47, size = 45, normalized size = 1.73

method	result	size
norman	\(x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \,{\mathrm e}^{5}+x^{3}+x^{2}}\right )^{2}\)	\(45\)
risch	\(x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \,{\mathrm e}^{5}+x^{3}+x^{2}}\right )^{2}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*ln(((-x+4)*exp(5)
-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*x^2)*ln
(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2*x^4-7*x
^3-5*x^2-x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.40, size = 113, normalized size = 4.35 \begin {gather*} x^{2} \log \left (-x^{3} + 3 \, x^{2} - x {\left (e^{5} - 4\right )} + 4 \, e^{5} + 1\right )^{2} + x^{2} \log \left (x^{2} + x + e^{5}\right )^{2} + 2 \, x^{2} \log \left (x^{2} + x + e^{5}\right ) \log \relax (x) + x^{2} \log \relax (x)^{2} - 2 \, {\left (x^{2} \log \left (x^{2} + x + e^{5}\right ) + x^{2} \log \relax (x)\right )} \log \left (-x^{3} + 3 \, x^{2} - x {\left (e^{5} - 4\right )} + 4 \, e^{5} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.12, size = 43, normalized size = 1.65 \begin {gather*} x^2\,{\ln \left (\frac {4\,x-{\mathrm {e}}^5\,\left (x-4\right )+3\,x^2-x^3+1}{x^3+x^2+{\mathrm {e}}^5\,x}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((4*x - exp(5)*(x - 4) + 3*x^2 - x^3 + 1)/(x*exp(5) + x^2 + x^3))^2*(exp(10)*(8*x - 2*x^2) + exp(5)*(2
*x + 16*x^2 + 12*x^3 - 4*x^4) + 2*x^2 + 10*x^3 + 14*x^4 + 4*x^5 - 2*x^6) - log((4*x - exp(5)*(x - 4) + 3*x^2 -
x^3 + 1)/(x*exp(5) + x^2 + x^3))*(8*x*exp(10) + exp(5)*(2*x + 16*x^2 + 16*x^3) + 4*x^2 + 14*x^3 + 16*x^4 + 8*
x^5))/(x + exp(5)*(8*x + 6*x^2 - 2*x^3 + 1) - exp(10)*(x - 4) + 5*x^2 + 7*x^3 + 2*x^4 - x^5),x)

[Out]

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

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sympy [A]  time = 0.50, size = 37, normalized size = 1.42 \begin {gather*} x^{2} \log {\left (\frac {- x^{3} + 3 x^{2} + 4 x + \left (4 - x\right ) e^{5} + 1}{x^{3} + x^{2} + x e^{5}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-8*x)*exp(5)**2+(4*x**4-12*x**3-16*x**2-2*x)*exp(5)+2*x**6-4*x**5-14*x**4-10*x**3-2*x**2)*l
n(((-x+4)*exp(5)-x**3+3*x**2+4*x+1)/(x*exp(5)+x**3+x**2))**2+(8*x*exp(5)**2+(16*x**3+16*x**2+2*x)*exp(5)+8*x**
5+16*x**4+14*x**3+4*x**2)*ln(((-x+4)*exp(5)-x**3+3*x**2+4*x+1)/(x*exp(5)+x**3+x**2)))/((x-4)*exp(5)**2+(2*x**3
-6*x**2-8*x-1)*exp(5)+x**5-2*x**4-7*x**3-5*x**2-x),x)

[Out]

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

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