3.58.25 \(\int \frac {40-44 x+12 x^2+e^{2 x^2} x^2+e^{x^2} (-13 x+7 x^2)+(15-10 x+3 x^2+e^{x^2} (x^2+6 x^3-2 x^4)) \log (\frac {x}{2})}{25-30 x+9 x^2+e^{2 x^2} x^2+e^{x^2} (-10 x+6 x^2)} \, dx\)

Optimal. Leaf size=27 \[ -2+x+\frac {(-3+x) \log \left (\frac {x}{2}\right )}{3+e^{x^2}-\frac {5}{x}} \]

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Rubi [F]  time = 1.89, 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 {40-44 x+12 x^2+e^{2 x^2} x^2+e^{x^2} \left (-13 x+7 x^2\right )+\left (15-10 x+3 x^2+e^{x^2} \left (x^2+6 x^3-2 x^4\right )\right ) \log \left (\frac {x}{2}\right )}{25-30 x+9 x^2+e^{2 x^2} x^2+e^{x^2} \left (-10 x+6 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(40 - 44*x + 12*x^2 + E^(2*x^2)*x^2 + E^x^2*(-13*x + 7*x^2) + (15 - 10*x + 3*x^2 + E^x^2*(x^2 + 6*x^3 - 2*
x^4))*Log[x/2])/(25 - 30*x + 9*x^2 + E^(2*x^2)*x^2 + E^x^2*(-10*x + 6*x^2)),x]

[Out]

x + 15*Log[x/2]*Defer[Int][(-5 + 3*x + E^x^2*x)^(-2), x] - 5*Log[x/2]*Defer[Int][x/(-5 + 3*x + E^x^2*x)^2, x]
+ 30*Log[x/2]*Defer[Int][x^2/(-5 + 3*x + E^x^2*x)^2, x] - 28*Log[x/2]*Defer[Int][x^3/(-5 + 3*x + E^x^2*x)^2, x
] + 6*Log[x/2]*Defer[Int][x^4/(-5 + 3*x + E^x^2*x)^2, x] - 3*Defer[Int][(-5 + 3*x + E^x^2*x)^(-1), x] + Defer[
Int][x/(-5 + 3*x + E^x^2*x), x] + Log[x/2]*Defer[Int][x/(-5 + 3*x + E^x^2*x), x] + 6*Log[x/2]*Defer[Int][x^2/(
-5 + 3*x + E^x^2*x), x] - 2*Log[x/2]*Defer[Int][x^3/(-5 + 3*x + E^x^2*x), x] - 15*Defer[Int][Defer[Int][(-5 +
(3 + E^x^2)*x)^(-2), x]/x, x] + 5*Defer[Int][Defer[Int][x/(-5 + (3 + E^x^2)*x)^2, x]/x, x] - 30*Defer[Int][Def
er[Int][x^2/(-5 + (3 + E^x^2)*x)^2, x]/x, x] + 28*Defer[Int][Defer[Int][x^3/(-5 + (3 + E^x^2)*x)^2, x]/x, x] -
6*Defer[Int][Defer[Int][x^4/(-5 + (3 + E^x^2)*x)^2, x]/x, x] - Defer[Int][Defer[Int][x/(-5 + (3 + E^x^2)*x),
x]/x, x] - 6*Defer[Int][Defer[Int][x^2/(-5 + (3 + E^x^2)*x), x]/x, x] + 2*Defer[Int][Defer[Int][x^3/(-5 + (3 +
E^x^2)*x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40-44 x+12 x^2+e^{2 x^2} x^2+e^{x^2} \left (-13 x+7 x^2\right )+\left (15-10 x+3 x^2+e^{x^2} \left (x^2+6 x^3-2 x^4\right )\right ) \log \left (\frac {x}{2}\right )}{\left (5-3 x-e^{x^2} x\right )^2} \, dx\\ &=\int \left (1+\frac {\left (15-5 x+30 x^2-28 x^3+6 x^4\right ) \log \left (\frac {x}{2}\right )}{\left (-5+3 x+e^{x^2} x\right )^2}-\frac {3-x-x \log \left (\frac {x}{2}\right )-6 x^2 \log \left (\frac {x}{2}\right )+2 x^3 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x}\right ) \, dx\\ &=x+\int \frac {\left (15-5 x+30 x^2-28 x^3+6 x^4\right ) \log \left (\frac {x}{2}\right )}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\int \frac {3-x-x \log \left (\frac {x}{2}\right )-6 x^2 \log \left (\frac {x}{2}\right )+2 x^3 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x} \, dx\\ &=x-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\int \frac {-3+x-x \left (-1-6 x+2 x^2\right ) \log \left (\frac {x}{2}\right )}{5-\left (3+e^{x^2}\right ) x} \, dx-\int \frac {15 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-5 \int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+30 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-28 \int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+6 \int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx\\ &=x-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\int \left (\frac {3}{-5+3 x+e^{x^2} x}-\frac {x}{-5+3 x+e^{x^2} x}-\frac {x \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x}-\frac {6 x^2 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x}+\frac {2 x^3 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x}\right ) \, dx-\int \left (\frac {15 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-5 \int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+30 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-28 \int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}+\frac {6 \int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}\right ) \, dx\\ &=x-2 \int \frac {x^3 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx+6 \int \frac {x^2 \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x} \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx+\int \frac {x \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x} \, dx-\int \frac {15 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-5 \int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+30 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-28 \int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \left (\frac {5 \left (3 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+6 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx\right )}{x}-\frac {28 \int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}\right ) \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx-5 \int \frac {3 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx+6 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx+28 \int \frac {\int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx-5 \int \left (\frac {3 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}+\frac {6 \int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}\right ) \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx+28 \int \frac {\int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx-5 \int \frac {3 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx-\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx+28 \int \frac {\int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-30 \int \frac {\int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx-5 \int \left (\frac {3 \int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}-\frac {\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x}\right ) \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx+28 \int \frac {\int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-30 \int \frac {\int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ &=x+2 \int \frac {\int \frac {x^3}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-3 \int \frac {1}{-5+3 x+e^{x^2} x} \, dx+5 \int \frac {\int \frac {x}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^4}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-6 \int \frac {\int \frac {x^2}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx-15 \int \frac {\int \frac {1}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+28 \int \frac {\int \frac {x^3}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx-30 \int \frac {\int \frac {x^2}{\left (-5+\left (3+e^{x^2}\right ) x\right )^2} \, dx}{x} \, dx+\log \left (\frac {x}{2}\right ) \int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\left (2 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{-5+3 x+e^{x^2} x} \, dx-\left (5 \log \left (\frac {x}{2}\right )\right ) \int \frac {x}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^4}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (6 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{-5+3 x+e^{x^2} x} \, dx+\left (15 \log \left (\frac {x}{2}\right )\right ) \int \frac {1}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx-\left (28 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^3}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\left (30 \log \left (\frac {x}{2}\right )\right ) \int \frac {x^2}{\left (-5+3 x+e^{x^2} x\right )^2} \, dx+\int \frac {x}{-5+3 x+e^{x^2} x} \, dx-\int \frac {\int \frac {x}{-5+\left (3+e^{x^2}\right ) x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 27, normalized size = 1.00 \begin {gather*} x+\frac {(-3+x) x \log \left (\frac {x}{2}\right )}{-5+3 x+e^{x^2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40 - 44*x + 12*x^2 + E^(2*x^2)*x^2 + E^x^2*(-13*x + 7*x^2) + (15 - 10*x + 3*x^2 + E^x^2*(x^2 + 6*x^
3 - 2*x^4))*Log[x/2])/(25 - 30*x + 9*x^2 + E^(2*x^2)*x^2 + E^x^2*(-10*x + 6*x^2)),x]

[Out]

x + ((-3 + x)*x*Log[x/2])/(-5 + 3*x + E^x^2*x)

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fricas [A]  time = 0.55, size = 43, normalized size = 1.59 \begin {gather*} \frac {x^{2} e^{\left (x^{2}\right )} + 3 \, x^{2} + {\left (x^{2} - 3 \, x\right )} \log \left (\frac {1}{2} \, x\right ) - 5 \, x}{x e^{\left (x^{2}\right )} + 3 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3+x^2)*exp(x^2)+3*x^2-10*x+15)*log(1/2*x)+x^2*exp(x^2)^2+(7*x^2-13*x)*exp(x^2)+12*x^2-
44*x+40)/(x^2*exp(x^2)^2+(6*x^2-10*x)*exp(x^2)+9*x^2-30*x+25),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 54, normalized size = 2.00 \begin {gather*} \frac {x^{2} e^{\left (x^{2}\right )} - x^{2} \log \relax (2) + x^{2} \log \relax (x) + 3 \, x^{2} + 3 \, x \log \relax (2) - 3 \, x \log \relax (x) - 5 \, x}{x e^{\left (x^{2}\right )} + 3 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3+x^2)*exp(x^2)+3*x^2-10*x+15)*log(1/2*x)+x^2*exp(x^2)^2+(7*x^2-13*x)*exp(x^2)+12*x^2-
44*x+40)/(x^2*exp(x^2)^2+(6*x^2-10*x)*exp(x^2)+9*x^2-30*x+25),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 25, normalized size = 0.93

method	result	size
risch	\(\frac {x \left (x -3\right ) \ln \left (\frac {x}{2}\right )}{{\mathrm e}^{x^{2}} x +3 x -5}+x\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4+6*x^3+x^2)*exp(x^2)+3*x^2-10*x+15)*ln(1/2*x)+x^2*exp(x^2)^2+(7*x^2-13*x)*exp(x^2)+12*x^2-44*x+40
)/(x^2*exp(x^2)^2+(6*x^2-10*x)*exp(x^2)+9*x^2-30*x+25),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 53, normalized size = 1.96 \begin {gather*} -\frac {x^{2} {\left (\log \relax (2) - 3\right )} - x^{2} e^{\left (x^{2}\right )} - x {\left (3 \, \log \relax (2) - 5\right )} - {\left (x^{2} - 3 \, x\right )} \log \relax (x)}{x e^{\left (x^{2}\right )} + 3 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3+x^2)*exp(x^2)+3*x^2-10*x+15)*log(1/2*x)+x^2*exp(x^2)^2+(7*x^2-13*x)*exp(x^2)+12*x^2-
44*x+40)/(x^2*exp(x^2)^2+(6*x^2-10*x)*exp(x^2)+9*x^2-30*x+25),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.71, size = 30, normalized size = 1.11 \begin {gather*} x-\frac {\ln \left (\frac {x}{2}\right )\,\left (3\,x-x^2\right )}{3\,x+x\,{\mathrm {e}}^{x^2}-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x/2)*(exp(x^2)*(x^2 + 6*x^3 - 2*x^4) - 10*x + 3*x^2 + 15) - exp(x^2)*(13*x - 7*x^2) - 44*x + x^2*exp(
2*x^2) + 12*x^2 + 40)/(x^2*exp(2*x^2) - exp(x^2)*(10*x - 6*x^2) - 30*x + 9*x^2 + 25),x)

[Out]

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

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sympy [A]  time = 0.30, size = 29, normalized size = 1.07 \begin {gather*} x + \frac {x^{2} \log {\left (\frac {x}{2} \right )} - 3 x \log {\left (\frac {x}{2} \right )}}{x e^{x^{2}} + 3 x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4+6*x**3+x**2)*exp(x**2)+3*x**2-10*x+15)*ln(1/2*x)+x**2*exp(x**2)**2+(7*x**2-13*x)*exp(x**2
)+12*x**2-44*x+40)/(x**2*exp(x**2)**2+(6*x**2-10*x)*exp(x**2)+9*x**2-30*x+25),x)

[Out]

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

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