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

Optimal. Leaf size=29 \[ \frac {1}{4} e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {-2+x}{5 x}\right ) \]

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Rubi [F]  time = 2.94, 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 {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, x^2/3] + (Sqrt[3*Pi]*Erfi[x/Sqrt[3]]*Log[1/5 - 2/(5*x)])/2 + 2*Log
[1/5 - 2/(5*x)]*Defer[Int][E^(x^2/3)/(-2 + x), x] - (Sqrt[3*Pi]*Defer[Int][Erfi[x/Sqrt[3]]/(-2 + x), x])/2 + (
14*Defer[Int][(E^(x^2/3)*Log[1/5 - 2/(5*x)]^2)/(2 - x), x])/3 + (14*Defer[Int][(E^(x^2/3)*Log[1/5 - 2/(5*x)]^2
)/(-2 + x), x])/3 + Defer[Int][E^(x^2/3)*x*Log[1/5 - 2/(5*x)]^2, x]/2 + Defer[Int][E^(x^2/3)*x^3*Log[1/5 - 2/(
5*x)]^2, x]/6 - 2*Defer[Int][Defer[Int][E^(x^2/3)/(-2 + x), x]/(-2 + x), x] + 2*Defer[Int][Defer[Int][E^(x^2/3
)/(-2 + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{\frac {x^2}{3}} x \log \left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+\frac {e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}+\frac {e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2 (-2+x)}+\frac {e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{3 (2-x)}+\frac {e^{\frac {x^2}{3}} x^4 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{6 (-2+x)}\right ) \, dx\\ &=\frac {1}{6} \int \frac {e^{\frac {x^2}{3}} x^4 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {1}{3} \int \frac {e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {1}{2} \int \frac {e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\int \frac {e^{\frac {x^2}{3}} x \log \left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\int \frac {e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx\\ &=\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int \left (8 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {16 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+4 e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+2 e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\frac {1}{3} \int \left (-4 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {8 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}-2 e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )-e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\frac {1}{2} \int \left (2 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {4 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx+\int \left (-e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {2 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}\right ) \, dx-\int \frac {-\sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right )-4 \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(2-x) x} \, dx\\ &=\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx-\int \left (\frac {\sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{(-2+x) x}+\frac {4 \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(-2+x) x}\right ) \, dx\\ &=\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-4 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(-2+x) x} \, dx-\sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{(-2+x) x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx\\ &=\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-4 \int \left (\frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{2 (-2+x)}-\frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{2 x}\right ) \, dx-\sqrt {3 \pi } \int \left (\frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{2 (-2+x)}-\frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{2 x}\right ) \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx\\ &=\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{-2+x} \, dx+2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{-2+x} \, dx+\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx\\ &=x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^2}{3}\right )+\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{-2+x} \, dx+2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{-2+x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.18, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.80, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, x^{2} e^{\left (\frac {1}{3} \, x^{2}\right )} \log \left (\frac {x - 2}{5 \, x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.33, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, x^{2} e^{\left (\frac {1}{3} \, x^{2}\right )} \log \left (\frac {x - 2}{5 \, x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.70, size = 820, normalized size = 28.28

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x^{2}}{3}} \ln \relax (5)^{2} x^{2}}{4}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} x^{2} \ln \relax (x )^{2}}{4}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} x^{2} \ln \left (x -2\right )^{2}}{4}+\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right ) \ln \relax (x )}{4}-\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \ln \relax (5) x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2}}{4}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} \ln \relax (5) x^{2} \ln \relax (x )}{2}+\left (-\frac {x^{2} {\mathrm e}^{\frac {x^{2}}{3}} \ln \relax (x )}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right ) {\mathrm e}^{\frac {x^{2}}{3}}}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2} {\mathrm e}^{\frac {x^{2}}{3}}}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2} {\mathrm e}^{\frac {x^{2}}{3}}}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{3} {\mathrm e}^{\frac {x^{2}}{3}}}{4}-\frac {\ln \relax (5) x^{2} {\mathrm e}^{\frac {x^{2}}{3}}}{2}\right ) \ln \left (x -2\right )-\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{6}}{16}-\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{4}}{16}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{5}}{8}-\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{4}}{16}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{5}}{8}-\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2}}{16}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{3}}{8}+\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{3}}{8}-\frac {{\mathrm e}^{\frac {x^{2}}{3}} \pi ^{2} x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{4}}{4}+\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \ln \relax (5) x^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{3}}{4}-\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \ln \relax (5) x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2}}{4}-\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2} \ln \relax (x )}{4}+\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{3} \ln \relax (x )}{4}+\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \ln \relax (5) x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )}{4}-\frac {i {\mathrm e}^{\frac {x^{2}}{3}} \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{x}\right )^{2} \ln \relax (x )}{4}\)	\(820\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(1/3*x^2)*ln(5)^2*x^2+1/4*exp(1/3*x^2)*x^2*ln(x)^2+1/4*exp(1/3*x^2)*x^2*ln(x-2)^2+1/4*I*exp(1/3*x^2)*Pi
*x^2*csgn(I*(x-2))*csgn(I/x)*csgn(I/x*(x-2))*ln(x)-1/4*I*exp(1/3*x^2)*Pi*ln(5)*x^2*csgn(I/x)*csgn(I/x*(x-2))^2
+1/2*exp(1/3*x^2)*ln(5)*x^2*ln(x)+(-1/2*x^2*exp(1/3*x^2)*ln(x)-1/4*I*Pi*x^2*csgn(I*(x-2))*csgn(I/x)*csgn(I/x*(
x-2))*exp(1/3*x^2)+1/4*I*Pi*x^2*csgn(I*(x-2))*csgn(I/x*(x-2))^2*exp(1/3*x^2)+1/4*I*Pi*x^2*csgn(I/x)*csgn(I/x*(
x-2))^2*exp(1/3*x^2)-1/4*I*Pi*x^2*csgn(I/x*(x-2))^3*exp(1/3*x^2)-1/2*ln(5)*x^2*exp(1/3*x^2))*ln(x-2)-1/16*exp(
1/3*x^2)*Pi^2*x^2*csgn(I/x*(x-2))^6-1/16*exp(1/3*x^2)*Pi^2*x^2*csgn(I*(x-2))^2*csgn(I/x*(x-2))^4+1/8*exp(1/3*x
^2)*Pi^2*x^2*csgn(I*(x-2))*csgn(I/x*(x-2))^5-1/16*exp(1/3*x^2)*Pi^2*x^2*csgn(I/x)^2*csgn(I/x*(x-2))^4+1/8*exp(
1/3*x^2)*Pi^2*x^2*csgn(I/x)*csgn(I/x*(x-2))^5-1/16*exp(1/3*x^2)*Pi^2*x^2*csgn(I*(x-2))^2*csgn(I/x)^2*csgn(I/x*
(x-2))^2+1/8*exp(1/3*x^2)*Pi^2*x^2*csgn(I*(x-2))^2*csgn(I/x)*csgn(I/x*(x-2))^3+1/8*exp(1/3*x^2)*Pi^2*x^2*csgn(
I*(x-2))*csgn(I/x)^2*csgn(I/x*(x-2))^3-1/4*exp(1/3*x^2)*Pi^2*x^2*csgn(I*(x-2))*csgn(I/x)*csgn(I/x*(x-2))^4+1/4
*I*exp(1/3*x^2)*Pi*ln(5)*x^2*csgn(I/x*(x-2))^3-1/4*I*exp(1/3*x^2)*Pi*ln(5)*x^2*csgn(I*(x-2))*csgn(I/x*(x-2))^2
-1/4*I*exp(1/3*x^2)*Pi*x^2*csgn(I*(x-2))*csgn(I/x*(x-2))^2*ln(x)+1/4*I*exp(1/3*x^2)*Pi*x^2*csgn(I/x*(x-2))^3*l
n(x)+1/4*I*exp(1/3*x^2)*Pi*ln(5)*x^2*csgn(I*(x-2))*csgn(I/x)*csgn(I/x*(x-2))-1/4*I*exp(1/3*x^2)*Pi*x^2*csgn(I/
x)*csgn(I/x*(x-2))^2*ln(x)

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maxima [B]  time = 0.52, size = 63, normalized size = 2.17 \begin {gather*} \frac {1}{4} \, {\left (x^{2} \log \relax (5)^{2} + x^{2} \log \left (x - 2\right )^{2} + 2 \, x^{2} \log \relax (5) \log \relax (x) + x^{2} \log \relax (x)^{2} - 2 \, {\left (x^{2} \log \relax (5) + x^{2} \log \relax (x)\right )} \log \left (x - 2\right )\right )} e^{\left (\frac {1}{3} \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.19, size = 22, normalized size = 0.76 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{\frac {x^2}{3}}\,{\ln \left (\frac {x-2}{5\,x}\right )}^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 71.02, size = 22, normalized size = 0.76 \begin {gather*} \frac {x^{2} e^{\frac {x^{2}}{3}} \log {\left (\frac {\frac {x}{5} - \frac {2}{5}}{x} \right )}^{2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4-2*x**3+3*x**2-6*x)*exp(1/3*x**2)*ln(1/5*(x-2)/x)**2+6*x*exp(1/3*x**2)*ln(1/5*(x-2)/x))/(6*x-1
2),x)

[Out]

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

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