3.58.23 \(\int \frac {8+8 x-6 x^2+4 x^3-5 x^4+2 x^5+e^9 (2+2 x-x^2)+(4 x^2-8 x^3+3 x^4) \log (\frac {x}{-2+x})+(-8 x+4 x^2+2 x^3-3 x^4+x^5+e^9 (-2 x+x^2)+(-8+e^9 (-2+x)+4 x+2 x^2-3 x^3+x^4) \log (\frac {x}{-2+x})) \log (\frac {4+e^9-x^2+x^3}{x+\log (\frac {x}{-2+x})})}{-24 x+12 x^2+6 x^3-9 x^4+3 x^5+e^9 (-6 x+3 x^2)+(-24+12 x+6 x^2-9 x^3+3 x^4+e^9 (-6+3 x)) \log (\frac {x}{-2+x})} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{3} x \log \left (\frac {4+e^9+x \left (-x+x^2\right )}{x+\log \left (\frac {x}{-2+x}\right )}\right ) \]

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Rubi [F]  time = 21.27, 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 {8+8 x-6 x^2+4 x^3-5 x^4+2 x^5+e^9 \left (2+2 x-x^2\right )+\left (4 x^2-8 x^3+3 x^4\right ) \log \left (\frac {x}{-2+x}\right )+\left (-8 x+4 x^2+2 x^3-3 x^4+x^5+e^9 \left (-2 x+x^2\right )+\left (-8+e^9 (-2+x)+4 x+2 x^2-3 x^3+x^4\right ) \log \left (\frac {x}{-2+x}\right )\right ) \log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right )}{-24 x+12 x^2+6 x^3-9 x^4+3 x^5+e^9 \left (-6 x+3 x^2\right )+\left (-24+12 x+6 x^2-9 x^3+3 x^4+e^9 (-6+3 x)\right ) \log \left (\frac {x}{-2+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 + 8*x - 6*x^2 + 4*x^3 - 5*x^4 + 2*x^5 + E^9*(2 + 2*x - x^2) + (4*x^2 - 8*x^3 + 3*x^4)*Log[x/(-2 + x)] +
(-8*x + 4*x^2 + 2*x^3 - 3*x^4 + x^5 + E^9*(-2*x + x^2) + (-8 + E^9*(-2 + x) + 4*x + 2*x^2 - 3*x^3 + x^4)*Log[
x/(-2 + x)])*Log[(4 + E^9 - x^2 + x^3)/(x + Log[x/(-2 + x)])])/(-24*x + 12*x^2 + 6*x^3 - 9*x^4 + 3*x^5 + E^9*(
-6*x + 3*x^2) + (-24 + 12*x + 6*x^2 - 9*x^3 + 3*x^4 + E^9*(-6 + 3*x))*Log[x/(-2 + x)]),x]

[Out]

(x*Log[(4 + E^9 - x^2 + x^3)/(x + Log[-(x/(2 - x))])])/3 - (2*Defer[Int][1/((-2 + x)*(x + Log[x/(-2 + x)])), x
])/3 + (16*Defer[Int][1/((-2 + x)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) + (2*E^9*Defer[Int][1/((-2 + x)*(x
+ Log[x/(-2 + x)])), x])/(3*(8 + E^9)) + ((4 + E^9)*Defer[Int][1/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])
), x])/3 - (16*Defer[Int][1/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) - (4*E^9*Defer[In
t][1/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) + (4*(4 + E^9)*Defer[Int][1/((4 + E^9 -
x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(8 + E^9) + (5*E^9*(4 + E^9)*Defer[Int][1/((4 + E^9 - x^2 + x^3)*(x + L
og[x/(-2 + x)])), x])/(3*(8 + E^9)) - (2*(32 + 20*E^9 + 3*E^18)*Defer[Int][1/((4 + E^9 - x^2 + x^3)*(x + Log[x
/(-2 + x)])), x])/(3*(8 + E^9)) + (4 + E^9)*Defer[Int][x/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x] - (
8*Defer[Int][x/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(8 + E^9) - (6*(4 + E^9)*Defer[Int][x/((4 +
E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(8 + E^9) - (2*E^9*(4 + E^9)*Defer[Int][x/((4 + E^9 - x^2 + x^3)*
(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) - (E^9*(10 + E^9)*Defer[Int][x/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2
+ x)])), x])/(3*(8 + E^9)) - Defer[Int][x^2/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x]/3 + (2*E^9*Defe
r[Int][x^2/((4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) - (5*(8 + 3*E^9)*Defer[Int][x^2/((
4 + E^9 - x^2 + x^3)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9)) + (2*(24 + 7*E^9)*Defer[Int][x^2/((4 + E^9 - x^
2 + x^3)*(x + Log[x/(-2 + x)])), x])/(3*(8 + E^9))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8-8 x+6 x^2-4 x^3+5 x^4-2 x^5-e^9 \left (2+2 x-x^2\right )-x^2 \left (4-8 x+3 x^2\right ) \log \left (\frac {x}{-2+x}\right )-\left (-8+e^9 (-2+x)+4 x+2 x^2-3 x^3+x^4\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right ) \log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right )}{3 \left (2 \left (4+e^9\right )-\left (4+e^9\right ) x-2 x^2+3 x^3-x^4\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {-8-8 x+6 x^2-4 x^3+5 x^4-2 x^5-e^9 \left (2+2 x-x^2\right )-x^2 \left (4-8 x+3 x^2\right ) \log \left (\frac {x}{-2+x}\right )-\left (-8+e^9 (-2+x)+4 x+2 x^2-3 x^3+x^4\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right ) \log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right )}{\left (2 \left (4+e^9\right )-\left (4+e^9\right ) x-2 x^2+3 x^3-x^4\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx\\ &=\frac {1}{3} \int \left (\frac {8}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}+\frac {8 x}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}-\frac {6 x^2}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}+\frac {4 x^3}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}-\frac {5 x^4}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}+\frac {2 x^5}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}-\frac {e^9 \left (-2-2 x+x^2\right )}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}+\frac {x^2 (-2+3 x) \log \left (\frac {x}{-2+x}\right )}{\left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )}+\log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right )\right ) \, dx\\ &=\frac {1}{3} \int \frac {x^2 (-2+3 x) \log \left (\frac {x}{-2+x}\right )}{\left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx+\frac {1}{3} \int \log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right ) \, dx+\frac {2}{3} \int \frac {x^5}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx+\frac {4}{3} \int \frac {x^3}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx-\frac {5}{3} \int \frac {x^4}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx-2 \int \frac {x^2}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx+\frac {8}{3} \int \frac {1}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx+\frac {8}{3} \int \frac {x}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx-\frac {1}{3} e^9 \int \frac {-2-2 x+x^2}{(-2+x) \left (4+e^9-x^2+x^3\right ) \left (x+\log \left (\frac {x}{-2+x}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 32, normalized size = 0.97 \begin {gather*} \frac {1}{3} x \log \left (\frac {4+e^9-x^2+x^3}{x+\log \left (\frac {x}{-2+x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 8*x - 6*x^2 + 4*x^3 - 5*x^4 + 2*x^5 + E^9*(2 + 2*x - x^2) + (4*x^2 - 8*x^3 + 3*x^4)*Log[x/(-2 +
x)] + (-8*x + 4*x^2 + 2*x^3 - 3*x^4 + x^5 + E^9*(-2*x + x^2) + (-8 + E^9*(-2 + x) + 4*x + 2*x^2 - 3*x^3 + x^4
)*Log[x/(-2 + x)])*Log[(4 + E^9 - x^2 + x^3)/(x + Log[x/(-2 + x)])])/(-24*x + 12*x^2 + 6*x^3 - 9*x^4 + 3*x^5 +
E^9*(-6*x + 3*x^2) + (-24 + 12*x + 6*x^2 - 9*x^3 + 3*x^4 + E^9*(-6 + 3*x))*Log[x/(-2 + x)]),x]

[Out]

(x*Log[(4 + E^9 - x^2 + x^3)/(x + Log[x/(-2 + x)])])/3

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fricas [A]  time = 0.99, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, x \log \left (\frac {x^{3} - x^{2} + e^{9} + 4}{x + \log \left (\frac {x}{x - 2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-2)*exp(9)+x^4-3*x^3+2*x^2+4*x-8)*log(x/(x-2))+(x^2-2*x)*exp(9)+x^5-3*x^4+2*x^3+4*x^2-8*x)*log(
(exp(9)+x^3-x^2+4)/(log(x/(x-2))+x))+(3*x^4-8*x^3+4*x^2)*log(x/(x-2))+(-x^2+2*x+2)*exp(9)+2*x^5-5*x^4+4*x^3-6*
x^2+8*x+8)/(((3*x-6)*exp(9)+3*x^4-9*x^3+6*x^2+12*x-24)*log(x/(x-2))+(3*x^2-6*x)*exp(9)+3*x^5-9*x^4+6*x^3+12*x^
2-24*x),x, algorithm="fricas")

[Out]

1/3*x*log((x^3 - x^2 + e^9 + 4)/(x + log(x/(x - 2))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{5} - 5 \, x^{4} + 4 \, x^{3} - 6 \, x^{2} - {\left (x^{2} - 2 \, x - 2\right )} e^{9} + {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 4 \, x^{2} + {\left (x^{2} - 2 \, x\right )} e^{9} + {\left (x^{4} - 3 \, x^{3} + 2 \, x^{2} + {\left (x - 2\right )} e^{9} + 4 \, x - 8\right )} \log \left (\frac {x}{x - 2}\right ) - 8 \, x\right )} \log \left (\frac {x^{3} - x^{2} + e^{9} + 4}{x + \log \left (\frac {x}{x - 2}\right )}\right ) + {\left (3 \, x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} \log \left (\frac {x}{x - 2}\right ) + 8 \, x + 8}{3 \, {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 4 \, x^{2} + {\left (x^{2} - 2 \, x\right )} e^{9} + {\left (x^{4} - 3 \, x^{3} + 2 \, x^{2} + {\left (x - 2\right )} e^{9} + 4 \, x - 8\right )} \log \left (\frac {x}{x - 2}\right ) - 8 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-2)*exp(9)+x^4-3*x^3+2*x^2+4*x-8)*log(x/(x-2))+(x^2-2*x)*exp(9)+x^5-3*x^4+2*x^3+4*x^2-8*x)*log(
(exp(9)+x^3-x^2+4)/(log(x/(x-2))+x))+(3*x^4-8*x^3+4*x^2)*log(x/(x-2))+(-x^2+2*x+2)*exp(9)+2*x^5-5*x^4+4*x^3-6*
x^2+8*x+8)/(((3*x-6)*exp(9)+3*x^4-9*x^3+6*x^2+12*x-24)*log(x/(x-2))+(3*x^2-6*x)*exp(9)+3*x^5-9*x^4+6*x^3+12*x^
2-24*x),x, algorithm="giac")

[Out]

integrate(1/3*(2*x^5 - 5*x^4 + 4*x^3 - 6*x^2 - (x^2 - 2*x - 2)*e^9 + (x^5 - 3*x^4 + 2*x^3 + 4*x^2 + (x^2 - 2*x
)*e^9 + (x^4 - 3*x^3 + 2*x^2 + (x - 2)*e^9 + 4*x - 8)*log(x/(x - 2)) - 8*x)*log((x^3 - x^2 + e^9 + 4)/(x + log
(x/(x - 2)))) + (3*x^4 - 8*x^3 + 4*x^2)*log(x/(x - 2)) + 8*x + 8)/(x^5 - 3*x^4 + 2*x^3 + 4*x^2 + (x^2 - 2*x)*e
^9 + (x^4 - 3*x^3 + 2*x^2 + (x - 2)*e^9 + 4*x - 8)*log(x/(x - 2)) - 8*x), x)

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maple [F]  time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (x -2\right ) {\mathrm e}^{9}+x^{4}-3 x^{3}+2 x^{2}+4 x -8\right ) \ln \left (\frac {x}{x -2}\right )+\left (x^{2}-2 x \right ) {\mathrm e}^{9}+x^{5}-3 x^{4}+2 x^{3}+4 x^{2}-8 x \right ) \ln \left (\frac {{\mathrm e}^{9}+x^{3}-x^{2}+4}{\ln \left (\frac {x}{x -2}\right )+x}\right )+\left (3 x^{4}-8 x^{3}+4 x^{2}\right ) \ln \left (\frac {x}{x -2}\right )+\left (-x^{2}+2 x +2\right ) {\mathrm e}^{9}+2 x^{5}-5 x^{4}+4 x^{3}-6 x^{2}+8 x +8}{\left (\left (3 x -6\right ) {\mathrm e}^{9}+3 x^{4}-9 x^{3}+6 x^{2}+12 x -24\right ) \ln \left (\frac {x}{x -2}\right )+\left (3 x^{2}-6 x \right ) {\mathrm e}^{9}+3 x^{5}-9 x^{4}+6 x^{3}+12 x^{2}-24 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-2)*exp(9)+x^4-3*x^3+2*x^2+4*x-8)*ln(x/(x-2))+(x^2-2*x)*exp(9)+x^5-3*x^4+2*x^3+4*x^2-8*x)*ln((exp(9)+
x^3-x^2+4)/(ln(x/(x-2))+x))+(3*x^4-8*x^3+4*x^2)*ln(x/(x-2))+(-x^2+2*x+2)*exp(9)+2*x^5-5*x^4+4*x^3-6*x^2+8*x+8)
/(((3*x-6)*exp(9)+3*x^4-9*x^3+6*x^2+12*x-24)*ln(x/(x-2))+(3*x^2-6*x)*exp(9)+3*x^5-9*x^4+6*x^3+12*x^2-24*x),x)

[Out]

int(((((x-2)*exp(9)+x^4-3*x^3+2*x^2+4*x-8)*ln(x/(x-2))+(x^2-2*x)*exp(9)+x^5-3*x^4+2*x^3+4*x^2-8*x)*ln((exp(9)+
x^3-x^2+4)/(ln(x/(x-2))+x))+(3*x^4-8*x^3+4*x^2)*ln(x/(x-2))+(-x^2+2*x+2)*exp(9)+2*x^5-5*x^4+4*x^3-6*x^2+8*x+8)
/(((3*x-6)*exp(9)+3*x^4-9*x^3+6*x^2+12*x-24)*ln(x/(x-2))+(3*x^2-6*x)*exp(9)+3*x^5-9*x^4+6*x^3+12*x^2-24*x),x)

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maxima [A]  time = 0.42, size = 31, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, x \log \left (x^{3} - x^{2} + e^{9} + 4\right ) - \frac {1}{3} \, x \log \left (x - \log \left (x - 2\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-2)*exp(9)+x^4-3*x^3+2*x^2+4*x-8)*log(x/(x-2))+(x^2-2*x)*exp(9)+x^5-3*x^4+2*x^3+4*x^2-8*x)*log(
(exp(9)+x^3-x^2+4)/(log(x/(x-2))+x))+(3*x^4-8*x^3+4*x^2)*log(x/(x-2))+(-x^2+2*x+2)*exp(9)+2*x^5-5*x^4+4*x^3-6*
x^2+8*x+8)/(((3*x-6)*exp(9)+3*x^4-9*x^3+6*x^2+12*x-24)*log(x/(x-2))+(3*x^2-6*x)*exp(9)+3*x^5-9*x^4+6*x^3+12*x^
2-24*x),x, algorithm="maxima")

[Out]

1/3*x*log(x^3 - x^2 + e^9 + 4) - 1/3*x*log(x - log(x - 2) + log(x))

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mupad [B]  time = 4.47, size = 29, normalized size = 0.88 \begin {gather*} \frac {x\,\ln \left (\frac {x^3-x^2+{\mathrm {e}}^9+4}{x+\ln \left (\frac {x}{x-2}\right )}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + exp(9)*(2*x - x^2 + 2) + log((exp(9) - x^2 + x^3 + 4)/(x + log(x/(x - 2))))*(4*x^2 - exp(9)*(2*x -
x^2) - 8*x + 2*x^3 - 3*x^4 + x^5 + log(x/(x - 2))*(4*x + exp(9)*(x - 2) + 2*x^2 - 3*x^3 + x^4 - 8)) - 6*x^2 +
4*x^3 - 5*x^4 + 2*x^5 + log(x/(x - 2))*(4*x^2 - 8*x^3 + 3*x^4) + 8)/(log(x/(x - 2))*(12*x + 6*x^2 - 9*x^3 + 3*
x^4 + exp(9)*(3*x - 6) - 24) - exp(9)*(6*x - 3*x^2) - 24*x + 12*x^2 + 6*x^3 - 9*x^4 + 3*x^5),x)

[Out]

(x*log((exp(9) - x^2 + x^3 + 4)/(x + log(x/(x - 2)))))/3

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-2)*exp(9)+x**4-3*x**3+2*x**2+4*x-8)*ln(x/(x-2))+(x**2-2*x)*exp(9)+x**5-3*x**4+2*x**3+4*x**2-8*
x)*ln((exp(9)+x**3-x**2+4)/(ln(x/(x-2))+x))+(3*x**4-8*x**3+4*x**2)*ln(x/(x-2))+(-x**2+2*x+2)*exp(9)+2*x**5-5*x
**4+4*x**3-6*x**2+8*x+8)/(((3*x-6)*exp(9)+3*x**4-9*x**3+6*x**2+12*x-24)*ln(x/(x-2))+(3*x**2-6*x)*exp(9)+3*x**5
-9*x**4+6*x**3+12*x**2-24*x),x)

[Out]

Exception raised: CoercionFailed

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