3.75.28 \(\int \frac {(16 x+4 e x+e^{3 x} (-18 x+54 x^2+e^3 (-2 x+6 x^2))) \log (\frac {-8-2 e+e^{3 x} (9+e^3)+9 x+e^3 x}{18 x+2 e^3 x})+(-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} (18 x+2 e^3 x)) \log ^2(\frac {-8-2 e+e^{3 x} (9+e^3)+9 x+e^3 x}{18 x+2 e^3 x})}{-8-2 e+e^{3 x} (9+e^3)+9 x+e^3 x} \, dx\)

Optimal. Leaf size=35 \[ x^2 \log ^2\left (\frac {-\frac {4+e}{9+e^3}+\frac {1}{2} \left (e^{3 x}+x\right )}{x}\right ) \]

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Rubi [F]  time = 10.04, 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 {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))*Log[(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x +
E^3*x)/(18*x + 2*E^3*x)] + (-16*x - 4*E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E + E^(
3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x),x]

[Out]

-1/2*x^2 + (5*x^3)/3 - (3*x^4)/2 + 2*x^3*Log[2*(9 + E^3)] + x^2*Log[2*(9 + E^3)]^2 - x^2*Log[9 + E^3 - (2*(4 +
E))/x + (E^(3*x)*(9 + E^3))/x] - 2*x^2*Log[2*(9 + E^3)]*Log[9 + E^3 - (2*(4 + E))/x + (E^(3*x)*(9 + E^3))/x]
+ 2*x^3*Log[-1/2*(2*(4 + E) - E^(3*x)*(9 + E^3) - (9 + E^3)*x)/((9 + E^3)*x)] - (33 + 6*E + E^3)*Defer[Int][x^
2/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] - 2*(33 + 6*E + E^3)*Log[2*(9 + E^3)]*Defer[Int]
[x^2/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] - 2*(33 + 6*E + E^3)*Log[-1/2*(2*(4 + E) - E^
(3*x)*(9 + E^3) - (9 + E^3)*x)/((9 + E^3)*x)]*Defer[Int][x^2/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3
/9)*x), x] + 3*(9 + E^3)*Defer[Int][x^3/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] + 2*(33 +
6*E + E^3)*Defer[Int][x^3/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] + 6*(9 + E^3)*Log[2*(9 +
E^3)]*Defer[Int][x^3/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] + 6*(9 + E^3)*Log[-1/2*(2*(4
+ E) - E^(3*x)*(9 + E^3) - (9 + E^3)*x)/((9 + E^3)*x)]*Defer[Int][x^3/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) -
9*(1 + E^3/9)*x), x] - 6*(9 + E^3)*Defer[Int][x^4/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x]
+ 2*Defer[Int][x*Log[9*(1 + E^3/9) - (8*(1 + E/4))/x + (9*E^(3*x)*(1 + E^3/9))/x]^2, x] + 6*(33 + 6*E + E^3)*D
efer[Int][Defer[Int][-(x^2/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/9)*x)), x], x] - 2*(33 + 6*E + E
^3)*Defer[Int][Defer[Int][-(x^2/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/9)*x)), x]/x, x] - 2*(33 +
6*E + E^3)^2*Defer[Int][Defer[Int][-(x^2/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/9)*x)), x]/(8*(1 +
E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] + 6*(9 + E^3)*(33 + 6*E + E^3)*Defer[Int][(x*Defer[Int][-
(x^2/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/9)*x)), x])/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(
1 + E^3/9)*x), x] - 18*(9 + E^3)*Defer[Int][Defer[Int][-(x^3/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^
3/9)*x)), x], x] + 6*(9 + E^3)*Defer[Int][Defer[Int][-(x^3/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/
9)*x)), x]/x, x] + 6*(9 + E^3)*(33 + 6*E + E^3)*Defer[Int][Defer[Int][-(x^3/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3
/9) + 9*(1 + E^3/9)*x)), x]/(8*(1 + E/4) - 9*E^(3*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x] - 18*(9 + E^3)^2*Defer
[Int][(x*Defer[Int][-(x^3/(-8*(1 + E/4) + 9*E^(3*x)*(1 + E^3/9) + 9*(1 + E^3/9)*x)), x])/(8*(1 + E/4) - 9*E^(3
*x)*(1 + E^3/9) - 9*(1 + E^3/9)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+\left (9+e^3\right ) x} \, dx\\ &=\int \frac {-\left (\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )\right )-\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{8 \left (1+\frac {e}{4}\right )-e^{3 x} \left (9+e^3\right )-\left (9+e^3\right ) x} \, dx\\ &=\int \left (\frac {2 x^2 \left (-33-6 e-e^3+3 \left (9+e^3\right ) x\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x}+2 x \left (-1+3 x+\log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \, dx\\ &=2 \int \frac {x^2 \left (-33-6 e-e^3+3 \left (9+e^3\right ) x\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x} \, dx+2 \int x \left (-1+3 x+\log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right ) \, dx\\ &=2 \int \left (x \left (\log \left (2 \left (9+e^3\right )\right )-\log \left (\frac {-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x}{x}\right )\right )+x \left (\log \left (2 \left (9+e^3\right )\right )-\log \left (\frac {-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x}{x}\right )\right )^2+3 x^2 \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+9 e^{3 x} \left (1+\frac {e^3}{9}\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \, dx-2 \int \frac {\left (8 \left (1+\frac {e}{4}\right )+9 e^{3 x} \left (1+\frac {e^3}{9}\right ) (-1+3 x)\right ) \left (\left (33+6 e+e^3\right ) \int -\frac {x^2}{-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x} \, dx-3 \left (9+e^3\right ) \int -\frac {x^3}{-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x} \, dx\right )}{x \left (8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x\right )} \, dx+\left (6 \left (9+e^3\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \int \frac {x^3}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x} \, dx-\left (2 \left (33+6 e+e^3\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \int \frac {x^2}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x} \, dx\\ &=2 \int x \left (\log \left (2 \left (9+e^3\right )\right )-\log \left (\frac {-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x}{x}\right )\right ) \, dx+2 \int x \left (\log \left (2 \left (9+e^3\right )\right )-\log \left (\frac {-8-2 e+9 e^{3 x}+e^{3+3 x}+9 x+e^3 x}{x}\right )\right )^2 \, dx-2 \int \left (\frac {(1-3 x) \left (-33 \left (1+\frac {1}{33} e \left (6+e^2\right )\right ) \int \frac {x^2}{-8-2 e+9 e^{3 x}+e^{3+3 x}+\left (9+e^3\right ) x} \, dx+27 \left (1+\frac {e^3}{9}\right ) \int \frac {x^3}{-8-2 e+9 e^{3 x}+e^{3+3 x}+\left (9+e^3\right ) x} \, dx\right )}{x}+\frac {\left (33+6 e+e^3-3 \left (9+e^3\right ) x\right ) \left (-33 \left (1+\frac {1}{33} e \left (6+e^2\right )\right ) \int \frac {x^2}{-8-2 e+9 e^{3 x}+e^{3+3 x}+\left (9+e^3\right ) x} \, dx+27 \left (1+\frac {e^3}{9}\right ) \int \frac {x^3}{-8-2 e+9 e^{3 x}+e^{3+3 x}+\left (9+e^3\right ) x} \, dx\right )}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x}\right ) \, dx+6 \int x^2 \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+9 e^{3 x} \left (1+\frac {e^3}{9}\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right ) \, dx+\left (6 \left (9+e^3\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \int \frac {x^3}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x} \, dx-\left (2 \left (33+6 e+e^3\right ) \log \left (\frac {-8 \left (1+\frac {e}{4}\right )+e^{3 x} \left (9+e^3\right )+9 \left (1+\frac {e^3}{9}\right ) x}{2 \left (9+e^3\right ) x}\right )\right ) \int \frac {x^2}{8 \left (1+\frac {e}{4}\right )-9 e^{3 x} \left (1+\frac {e^3}{9}\right )-9 \left (1+\frac {e^3}{9}\right ) x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (16 x+4 e x+e^{3 x} \left (-18 x+54 x^2+e^3 \left (-2 x+6 x^2\right )\right )\right ) \log \left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )+\left (-16 x-4 e x+18 x^2+2 e^3 x^2+e^{3 x} \left (18 x+2 e^3 x\right )\right ) \log ^2\left (\frac {-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x}{18 x+2 e^3 x}\right )}{-8-2 e+e^{3 x} \left (9+e^3\right )+9 x+e^3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))*Log[(-8 - 2*E + E^(3*x)*(9 + E^3) +
9*x + E^3*x)/(18*x + 2*E^3*x)] + (-16*x - 4*E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E
+ E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x),x]

[Out]

Integrate[((16*x + 4*E*x + E^(3*x)*(-18*x + 54*x^2 + E^3*(-2*x + 6*x^2)))*Log[(-8 - 2*E + E^(3*x)*(9 + E^3) +
9*x + E^3*x)/(18*x + 2*E^3*x)] + (-16*x - 4*E*x + 18*x^2 + 2*E^3*x^2 + E^(3*x)*(18*x + 2*E^3*x))*Log[(-8 - 2*E
+ E^(3*x)*(9 + E^3) + 9*x + E^3*x)/(18*x + 2*E^3*x)]^2)/(-8 - 2*E + E^(3*x)*(9 + E^3) + 9*x + E^3*x), x]

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fricas [A]  time = 0.70, size = 41, normalized size = 1.17 \begin {gather*} x^{2} \log \left (\frac {x e^{3} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} + 9 \, x - 2 \, e - 8}{2 \, {\left (x e^{3} + 9 \, x\right )}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*x*exp(1)+18*x^2-16*x)*log(((exp(3)+9)*exp(3*x)+x*exp(3)-
2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*log(((exp(3)+
9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8),x, algo
rithm="fricas")

[Out]

x^2*log(1/2*(x*e^3 + (e^3 + 9)*e^(3*x) + 9*x - 2*e - 8)/(x*e^3 + 9*x))^2

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giac [B]  time = 9.38, size = 90, normalized size = 2.57 \begin {gather*} x^{2} \log \left (2 \, x e^{3} + 18 \, x\right )^{2} - 2 \, x^{2} \log \left (2 \, x e^{3} + 18 \, x\right ) \log \left (x e^{3} + 9 \, x - 2 \, e + 9 \, e^{\left (3 \, x\right )} + e^{\left (3 \, x + 3\right )} - 8\right ) + x^{2} \log \left (x e^{3} + 9 \, x - 2 \, e + 9 \, e^{\left (3 \, x\right )} + e^{\left (3 \, x + 3\right )} - 8\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*x*exp(1)+18*x^2-16*x)*log(((exp(3)+9)*exp(3*x)+x*exp(3)-
2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*log(((exp(3)+
9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8),x, algo
rithm="giac")

[Out]

x^2*log(2*x*e^3 + 18*x)^2 - 2*x^2*log(2*x*e^3 + 18*x)*log(x*e^3 + 9*x - 2*e + 9*e^(3*x) + e^(3*x + 3) - 8) + x
^2*log(x*e^3 + 9*x - 2*e + 9*e^(3*x) + e^(3*x + 3) - 8)^2

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maple [A]  time = 0.65, size = 42, normalized size = 1.20

method	result	size
norman	\(x^{2} \ln \left (\frac {\left ({\mathrm e}^{3}+9\right ) {\mathrm e}^{3 x}+x \,{\mathrm e}^{3}-2 \,{\mathrm e}+9 x -8}{2 x \,{\mathrm e}^{3}+18 x}\right )^{2}\)	\(42\)
risch	\(\text {Expression too large to display}\)	\(2142\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*x*exp(1)+18*x^2-16*x)*ln(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1
)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*ln(((exp(3)+9)*exp(3
*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8),x,method=_RETUR
NVERBOSE)

[Out]

x^2*ln(((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2

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maxima [B]  time = 0.52, size = 120, normalized size = 3.43 \begin {gather*} x^{2} \log \left (x {\left (e^{3} + 9\right )} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} - 2 \, e - 8\right )^{2} + 2 \, x^{2} {\left (\log \relax (2) + \log \left (e^{3} + 9\right )\right )} \log \relax (x) + x^{2} \log \relax (x)^{2} + {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (e^{3} + 9\right ) + \log \left (e^{3} + 9\right )^{2}\right )} x^{2} - 2 \, {\left (x^{2} {\left (\log \relax (2) + \log \left (e^{3} + 9\right )\right )} + x^{2} \log \relax (x)\right )} \log \left (x {\left (e^{3} + 9\right )} + {\left (e^{3} + 9\right )} e^{\left (3 \, x\right )} - 2 \, e - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x^2*exp(3)-4*x*exp(1)+18*x^2-16*x)*log(((exp(3)+9)*exp(3*x)+x*exp(3)-
2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))^2+(((6*x^2-2*x)*exp(3)+54*x^2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*log(((exp(3)+
9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8),x, algo
rithm="maxima")

[Out]

x^2*log(x*(e^3 + 9) + (e^3 + 9)*e^(3*x) - 2*e - 8)^2 + 2*x^2*(log(2) + log(e^3 + 9))*log(x) + x^2*log(x)^2 + (
log(2)^2 + 2*log(2)*log(e^3 + 9) + log(e^3 + 9)^2)*x^2 - 2*(x^2*(log(2) + log(e^3 + 9)) + x^2*log(x))*log(x*(e
^3 + 9) + (e^3 + 9)*e^(3*x) - 2*e - 8)

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mupad [B]  time = 6.27, size = 41, normalized size = 1.17 \begin {gather*} x^2\,{\ln \left (\frac {9\,x-2\,\mathrm {e}+x\,{\mathrm {e}}^3+{\mathrm {e}}^{3\,x}\,\left ({\mathrm {e}}^3+9\right )-8}{18\,x+2\,x\,{\mathrm {e}}^3}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(18*x + 2*x*exp(3)))*(16*x - exp(3*x)*(18*x +
exp(3)*(2*x - 6*x^2) - 54*x^2) + 4*x*exp(1)) + log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(1
8*x + 2*x*exp(3)))^2*(exp(3*x)*(18*x + 2*x*exp(3)) - 16*x - 4*x*exp(1) + 2*x^2*exp(3) + 18*x^2))/(9*x - 2*exp(
1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8),x)

[Out]

x^2*log((9*x - 2*exp(1) + x*exp(3) + exp(3*x)*(exp(3) + 9) - 8)/(18*x + 2*x*exp(3)))^2

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sympy [A]  time = 0.75, size = 41, normalized size = 1.17 \begin {gather*} x^{2} \log {\left (\frac {9 x + x e^{3} + \left (9 + e^{3}\right ) e^{3 x} - 8 - 2 e}{18 x + 2 x e^{3}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*exp(3)+18*x)*exp(3*x)+2*x**2*exp(3)-4*x*exp(1)+18*x**2-16*x)*ln(((exp(3)+9)*exp(3*x)+x*exp(3)
-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x))**2+(((6*x**2-2*x)*exp(3)+54*x**2-18*x)*exp(3*x)+4*x*exp(1)+16*x)*ln(((exp(
3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8)/(2*x*exp(3)+18*x)))/((exp(3)+9)*exp(3*x)+x*exp(3)-2*exp(1)+9*x-8),x)

[Out]

x**2*log((9*x + x*exp(3) + (9 + exp(3))*exp(3*x) - 8 - 2*E)/(18*x + 2*x*exp(3)))**2

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