3.29.64 \(\int \frac {-42+10 x+36 x^2-9 x^4-2 x^5+(45+27 x-29 x^2-17 x^3+x^4+x^5) \log (\frac {e^{2 x}}{3+x})}{-3 x-x^2+(27-9 x-21 x^2+x^3+5 x^4+x^5) \log (\frac {e^{2 x}}{3+x})} \, dx\)

Optimal. Leaf size=30 \[ -4+x-\log \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right ) \]

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Rubi [F]  time = 3.29, 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 {-42+10 x+36 x^2-9 x^4-2 x^5+\left (45+27 x-29 x^2-17 x^3+x^4+x^5\right ) \log \left (\frac {e^{2 x}}{3+x}\right )}{-3 x-x^2+\left (27-9 x-21 x^2+x^3+5 x^4+x^5\right ) \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-42 + 10*x + 36*x^2 - 9*x^4 - 2*x^5 + (45 + 27*x - 29*x^2 - 17*x^3 + x^4 + x^5)*Log[E^(2*x)/(3 + x)])/(-3
*x - x^2 + (27 - 9*x - 21*x^2 + x^3 + 5*x^4 + x^5)*Log[E^(2*x)/(3 + x)]),x]

[Out]

x - 2*Log[3 - x - x^2] + (2*(13 + Sqrt[13])*Defer[Int][1/((-1 - Sqrt[13] - 2*x)*(x - (-3 + x + x^2)^2*Log[E^(2
*x)/(3 + x)])), x])/13 - (24*Defer[Int][1/((-1 + Sqrt[13] - 2*x)*(x - (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)])),
x])/Sqrt[13] + (2*(13 - Sqrt[13])*Defer[Int][1/((-1 + Sqrt[13] - 2*x)*(x - (-3 + x + x^2)^2*Log[E^(2*x)/(3 +
x)])), x])/13 + 9*Defer[Int][1/((-3 - x)*(x - (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)])), x] - (24*Defer[Int][1/(
(1 + Sqrt[13] + 2*x)*(x - (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)])), x])/Sqrt[13] - 21*Defer[Int][(-x + (-3 + x
+ x^2)^2*Log[E^(2*x)/(3 + x)])^(-1), x] + 10*Defer[Int][x/(-x + (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)]), x] + 9
*Defer[Int][x^2/(-x + (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)]), x] - 3*Defer[Int][x^3/(-x + (-3 + x + x^2)^2*Log
[E^(2*x)/(3 + x)]), x] - 2*Defer[Int][x^4/(-x + (-3 + x + x^2)^2*Log[E^(2*x)/(3 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-5-3 x+x^2}{-3+x+x^2}+\frac {126-87 x-154 x^2+46 x^3+64 x^4-3 x^5-11 x^6-2 x^7}{(3+x) \left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}\right ) \, dx\\ &=\int \frac {-5-3 x+x^2}{-3+x+x^2} \, dx+\int \frac {126-87 x-154 x^2+46 x^3+64 x^4-3 x^5-11 x^6-2 x^7}{(3+x) \left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx\\ &=\int \left (1-\frac {2 (1+2 x)}{-3+x+x^2}\right ) \, dx+\int \frac {126-87 x-154 x^2+46 x^3+64 x^4-3 x^5-11 x^6-2 x^7}{(3+x) \left (3-x-x^2\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx\\ &=x-2 \int \frac {1+2 x}{-3+x+x^2} \, dx+\int \left (-\frac {21}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )}+\frac {10 x}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )}+\frac {9 x^2}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )}-\frac {3 x^3}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )}-\frac {2 x^4}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )}+\frac {9}{(3+x) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}+\frac {2 (-6+x)}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}\right ) \, dx\\ &=x-2 \log \left (3-x-x^2\right )-2 \int \frac {x^4}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+2 \int \frac {-6+x}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-3 \int \frac {x^3}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {x^2}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(3+x) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+10 \int \frac {x}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-21 \int \frac {1}{-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )+2 \int \frac {-6+x}{\left (3-x-x^2\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+2 \int \left (-\frac {6}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}+\frac {x}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}\right ) \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )+2 \int \frac {x}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-12 \int \frac {1}{\left (-3+x+x^2\right ) \left (-x+9 \log \left (\frac {e^{2 x}}{3+x}\right )-6 x \log \left (\frac {e^{2 x}}{3+x}\right )-5 x^2 \log \left (\frac {e^{2 x}}{3+x}\right )+2 x^3 \log \left (\frac {e^{2 x}}{3+x}\right )+x^4 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )+2 \int \frac {x}{\left (3-x-x^2\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-12 \int \frac {1}{\left (3-x-x^2\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+2 \int \left (\frac {1+\frac {1}{\sqrt {13}}}{\left (-1-\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}+\frac {1-\frac {1}{\sqrt {13}}}{\left (-1+\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}\right ) \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-12 \int \left (\frac {2}{\sqrt {13} \left (-1+\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}+\frac {2}{\sqrt {13} \left (1+\sqrt {13}+2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )}\right ) \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx\\ &=x-2 \log \left (3-x-x^2\right )-2 \int \frac {x^4}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-3 \int \frac {x^3}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+9 \int \frac {1}{(-3-x) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+9 \int \frac {x^2}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx+10 \int \frac {x}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-21 \int \frac {1}{-x+\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )} \, dx-\frac {24 \int \frac {1}{\left (-1+\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx}{\sqrt {13}}-\frac {24 \int \frac {1}{\left (1+\sqrt {13}+2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx}{\sqrt {13}}+\frac {1}{13} \left (2 \left (13-\sqrt {13}\right )\right ) \int \frac {1}{\left (-1+\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx+\frac {1}{13} \left (2 \left (13+\sqrt {13}\right )\right ) \int \frac {1}{\left (-1-\sqrt {13}-2 x\right ) \left (x-\left (-3+x+x^2\right )^2 \log \left (\frac {e^{2 x}}{3+x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 217, normalized size = 7.23 \begin {gather*} x-\log \left (17 x-12 x^2-10 x^3+4 x^4+2 x^5+9 \log \left (\frac {1}{3+x}\right )-6 x \log \left (\frac {1}{3+x}\right )-5 x^2 \log \left (\frac {1}{3+x}\right )+2 x^3 \log \left (\frac {1}{3+x}\right )+x^4 \log \left (\frac {1}{3+x}\right )+9 \left (-2 x-\log \left (\frac {1}{3+x}\right )+\log \left (\frac {e^{2 x}}{3+x}\right )\right )-6 x \left (-2 x-\log \left (\frac {1}{3+x}\right )+\log \left (\frac {e^{2 x}}{3+x}\right )\right )-5 x^2 \left (-2 x-\log \left (\frac {1}{3+x}\right )+\log \left (\frac {e^{2 x}}{3+x}\right )\right )+2 x^3 \left (-2 x-\log \left (\frac {1}{3+x}\right )+\log \left (\frac {e^{2 x}}{3+x}\right )\right )+x^4 \left (-2 x-\log \left (\frac {1}{3+x}\right )+\log \left (\frac {e^{2 x}}{3+x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-42 + 10*x + 36*x^2 - 9*x^4 - 2*x^5 + (45 + 27*x - 29*x^2 - 17*x^3 + x^4 + x^5)*Log[E^(2*x)/(3 + x)
])/(-3*x - x^2 + (27 - 9*x - 21*x^2 + x^3 + 5*x^4 + x^5)*Log[E^(2*x)/(3 + x)]),x]

[Out]

x - Log[17*x - 12*x^2 - 10*x^3 + 4*x^4 + 2*x^5 + 9*Log[(3 + x)^(-1)] - 6*x*Log[(3 + x)^(-1)] - 5*x^2*Log[(3 +
x)^(-1)] + 2*x^3*Log[(3 + x)^(-1)] + x^4*Log[(3 + x)^(-1)] + 9*(-2*x - Log[(3 + x)^(-1)] + Log[E^(2*x)/(3 + x)
]) - 6*x*(-2*x - Log[(3 + x)^(-1)] + Log[E^(2*x)/(3 + x)]) - 5*x^2*(-2*x - Log[(3 + x)^(-1)] + Log[E^(2*x)/(3
+ x)]) + 2*x^3*(-2*x - Log[(3 + x)^(-1)] + Log[E^(2*x)/(3 + x)]) + x^4*(-2*x - Log[(3 + x)^(-1)] + Log[E^(2*x)
/(3 + x)])]

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fricas [B]  time = 0.55, size = 69, normalized size = 2.30 \begin {gather*} x - 2 \, \log \left (x^{2} + x - 3\right ) - \log \left (\frac {{\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x + 3}\right ) - x}{x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+x^4-17*x^3-29*x^2+27*x+45)*log(exp(2*x)/(3+x))-2*x^5-9*x^4+36*x^2+10*x-42)/((x^5+5*x^4+x^3-21*
x^2-9*x+27)*log(exp(2*x)/(3+x))-x^2-3*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.23, size = 68, normalized size = 2.27 \begin {gather*} x - \log \left (-2 \, x^{5} + x^{4} \log \left (x + 3\right ) - 4 \, x^{4} + 2 \, x^{3} \log \left (x + 3\right ) + 10 \, x^{3} - 5 \, x^{2} \log \left (x + 3\right ) + 12 \, x^{2} - 6 \, x \log \left (x + 3\right ) - 17 \, x + 9 \, \log \left (x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+x^4-17*x^3-29*x^2+27*x+45)*log(exp(2*x)/(3+x))-2*x^5-9*x^4+36*x^2+10*x-42)/((x^5+5*x^4+x^3-21*
x^2-9*x+27)*log(exp(2*x)/(3+x))-x^2-3*x),x, algorithm="giac")

[Out]

x - log(-2*x^5 + x^4*log(x + 3) - 4*x^4 + 2*x^3*log(x + 3) + 10*x^3 - 5*x^2*log(x + 3) + 12*x^2 - 6*x*log(x +
3) - 17*x + 9*log(x + 3))

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maple [B]  time = 0.29, size = 170, normalized size = 5.67

method	result	size
default	\(x -\ln \left (2 x^{5}-\ln \left (3+x \right ) x^{4}+x^{4} \left (\ln \left (\frac {{\mathrm e}^{2 x}}{3+x}\right )-2 x +\ln \left (3+x \right )\right )+4 x^{4}-2 \ln \left (3+x \right ) x^{3}+2 x^{3} \left (\ln \left (\frac {{\mathrm e}^{2 x}}{3+x}\right )-2 x +\ln \left (3+x \right )\right )-10 x^{3}+5 \ln \left (3+x \right ) x^{2}-5 x^{2} \left (\ln \left (\frac {{\mathrm e}^{2 x}}{3+x}\right )-2 x +\ln \left (3+x \right )\right )-12 x^{2}+6 x \ln \left (3+x \right )-6 x \left (\ln \left (\frac {{\mathrm e}^{2 x}}{3+x}\right )-2 x +\ln \left (3+x \right )\right )-x +9 \ln \left (\frac {{\mathrm e}^{2 x}}{3+x}\right )\right )\)	\(170\)
risch	\(x -2 \ln \left (x^{2}+x -3\right )-\ln \left (\ln \left ({\mathrm e}^{2 x}\right )-\frac {i \left (-2 i x -2 \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+5 \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+9 \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )+6 \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}-\pi \,x^{4} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}-2 \pi \,x^{3} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+5 \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+6 \pi x \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}-\pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+2 \pi \,x^{3} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )-5 \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )-6 \pi x \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )+\pi \,x^{4} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )-18 i \ln \left (3+x \right )+9 \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{3}-9 \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}-9 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{2}+\pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{3}+10 i x^{2} \ln \left (3+x \right )+12 i x \ln \left (3+x \right )-2 i x^{4} \ln \left (3+x \right )-4 i x^{3} \ln \left (3+x \right )+2 \pi \,x^{3} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{3}-5 \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{3}-6 \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{3+x}\right )^{3}\right )}{2 \left (x^{4}+2 x^{3}-5 x^{2}-6 x +9\right )}\right )\)	\(652\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5+x^4-17*x^3-29*x^2+27*x+45)*ln(exp(2*x)/(3+x))-2*x^5-9*x^4+36*x^2+10*x-42)/((x^5+5*x^4+x^3-21*x^2-9*x
+27)*ln(exp(2*x)/(3+x))-x^2-3*x),x,method=_RETURNVERBOSE)

[Out]

x-ln(2*x^5-ln(3+x)*x^4+x^4*(ln(exp(2*x)/(3+x))-2*x+ln(3+x))+4*x^4-2*ln(3+x)*x^3+2*x^3*(ln(exp(2*x)/(3+x))-2*x+
ln(3+x))-10*x^3+5*ln(3+x)*x^2-5*x^2*(ln(exp(2*x)/(3+x))-2*x+ln(3+x))-12*x^2+6*x*ln(3+x)-6*x*(ln(exp(2*x)/(3+x)
)-2*x+ln(3+x))-x+9*ln(exp(2*x)/(3+x)))

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maxima [B]  time = 0.75, size = 84, normalized size = 2.80 \begin {gather*} x - 2 \, \log \left (x^{2} + x - 3\right ) - \log \left (-\frac {2 \, x^{5} + 4 \, x^{4} - 10 \, x^{3} - 12 \, x^{2} - {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )} \log \left (x + 3\right ) + 17 \, x}{x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+x^4-17*x^3-29*x^2+27*x+45)*log(exp(2*x)/(3+x))-2*x^5-9*x^4+36*x^2+10*x-42)/((x^5+5*x^4+x^3-21*
x^2-9*x+27)*log(exp(2*x)/(3+x))-x^2-3*x),x, algorithm="maxima")

[Out]

x - 2*log(x^2 + x - 3) - log(-(2*x^5 + 4*x^4 - 10*x^3 - 12*x^2 - (x^4 + 2*x^3 - 5*x^2 - 6*x + 9)*log(x + 3) +
17*x)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {10\,x+36\,x^2-9\,x^4-2\,x^5+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x+3}\right )\,\left (x^5+x^4-17\,x^3-29\,x^2+27\,x+45\right )-42}{3\,x+x^2-\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x+3}\right )\,\left (x^5+5\,x^4+x^3-21\,x^2-9\,x+27\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x + 36*x^2 - 9*x^4 - 2*x^5 + log(exp(2*x)/(x + 3))*(27*x - 29*x^2 - 17*x^3 + x^4 + x^5 + 45) - 42)/(3
*x + x^2 - log(exp(2*x)/(x + 3))*(x^3 - 21*x^2 - 9*x + 5*x^4 + x^5 + 27)),x)

[Out]

-int((10*x + 36*x^2 - 9*x^4 - 2*x^5 + log(exp(2*x)/(x + 3))*(27*x - 29*x^2 - 17*x^3 + x^4 + x^5 + 45) - 42)/(3
*x + x^2 - log(exp(2*x)/(x + 3))*(x^3 - 21*x^2 - 9*x + 5*x^4 + x^5 + 27)), x)

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sympy [A]  time = 0.75, size = 42, normalized size = 1.40 \begin {gather*} x - \log {\left (- \frac {x}{x^{4} + 2 x^{3} - 5 x^{2} - 6 x + 9} + \log {\left (\frac {e^{2 x}}{x + 3} \right )} \right )} - 2 \log {\left (x^{2} + x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5+x**4-17*x**3-29*x**2+27*x+45)*ln(exp(2*x)/(3+x))-2*x**5-9*x**4+36*x**2+10*x-42)/((x**5+5*x**4
+x**3-21*x**2-9*x+27)*ln(exp(2*x)/(3+x))-x**2-3*x),x)

[Out]

x - log(-x/(x**4 + 2*x**3 - 5*x**2 - 6*x + 9) + log(exp(2*x)/(x + 3))) - 2*log(x**2 + x - 3)

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