3.39.72 \(\int \frac {(-12+24 x-12 x^2-12 x^3-2 x^4) \log (e^{2 x} x^2)+(-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5) \log (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2})}{(6 x-15 x^2+4 x^3+6 x^4+x^5) \log (e^{2 x} x^2) \log (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2})} \, dx\)

Optimal. Leaf size=31 \[ \log \left (\frac {1}{\log \left (e^{2 x} x^2\right ) \log ^2\left (-x+\frac {1}{3-\frac {2}{x}+x}\right )}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 2.65, 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 (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-12 + 24*x - 12*x^2 - 12*x^3 - 2*x^4)*Log[E^(2*x)*x^2] + (-12 + 18*x + 22*x^2 - 20*x^3 - 14*x^4 - 2*x^5)
*Log[(3*x - 3*x^2 - x^3)/(-2 + 3*x + x^2)])/((6*x - 15*x^2 + 4*x^3 + 6*x^4 + x^5)*Log[E^(2*x)*x^2]*Log[(3*x -
3*x^2 - x^3)/(-2 + 3*x + x^2)]),x]

[Out]

-Log[Log[E^(2*x)*x^2]] - (12*Defer[Int][1/((-3 + Sqrt[17] - 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]
), x])/Sqrt[17] + 4*Sqrt[3/7]*Defer[Int][1/((-3 + Sqrt[21] - 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))
]), x] - 2*Defer[Int][1/(x*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x] + (4*(17 - 3*Sqrt[17])*Defer[Int
][1/((3 - Sqrt[17] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x])/17 - (12*Defer[Int][1/((3 + Sqrt
[17] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x])/Sqrt[17] + (4*(17 + 3*Sqrt[17])*Defer[Int][1/(
(3 + Sqrt[17] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x])/17 - (4*(7 - Sqrt[21])*Defer[Int][1/(
(3 - Sqrt[21] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x])/7 + 4*Sqrt[3/7]*Defer[Int][1/((3 + Sq
rt[21] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x] - (4*(7 + Sqrt[21])*Defer[Int][1/((3 + Sqrt[2
1] + 2*x)*Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]), x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-\frac {6-9 x-11 x^2+10 x^3+7 x^4+x^5}{\log \left (e^{2 x} x^2\right )}-\frac {6-12 x+6 x^2+6 x^3+x^4}{\log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right )}{x \left (6-15 x+4 x^2+6 x^3+x^4\right )} \, dx\\ &=2 \int \frac {-\frac {6-9 x-11 x^2+10 x^3+7 x^4+x^5}{\log \left (e^{2 x} x^2\right )}-\frac {6-12 x+6 x^2+6 x^3+x^4}{\log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}}{x \left (6-15 x+4 x^2+6 x^3+x^4\right )} \, dx\\ &=2 \int \left (\frac {-1-x}{x \log \left (e^{2 x} x^2\right )}+\frac {-6+12 x-6 x^2-6 x^3-x^4}{x \left (-3+3 x+x^2\right ) \left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx\\ &=2 \int \frac {-1-x}{x \log \left (e^{2 x} x^2\right )} \, dx+2 \int \frac {-6+12 x-6 x^2-6 x^3-x^4}{x \left (-3+3 x+x^2\right ) \left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )+2 \int \left (-\frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}+\frac {-3-2 x}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}+\frac {3+2 x}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )-2 \int \frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+2 \int \frac {-3-2 x}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+2 \int \frac {3+2 x}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )+2 \int \left (-\frac {3}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}-\frac {2 x}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx+2 \int \left (\frac {3}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}+\frac {2 x}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx-2 \int \frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )-2 \int \frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx-4 \int \frac {x}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+4 \int \frac {x}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx-6 \int \frac {1}{\left (-3+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+6 \int \frac {1}{\left (-2+3 x+x^2\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )-2 \int \frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+4 \int \left (\frac {1-\frac {3}{\sqrt {17}}}{\left (3-\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}+\frac {1+\frac {3}{\sqrt {17}}}{\left (3+\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx-4 \int \left (\frac {1-\sqrt {\frac {3}{7}}}{\left (3-\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}+\frac {1+\sqrt {\frac {3}{7}}}{\left (3+\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx+6 \int \left (-\frac {2}{\sqrt {17} \left (-3+\sqrt {17}-2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}-\frac {2}{\sqrt {17} \left (3+\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx-6 \int \left (-\frac {2}{\sqrt {21} \left (-3+\sqrt {21}-2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}-\frac {2}{\sqrt {21} \left (3+\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )}\right ) \, dx\\ &=-\log \left (\log \left (e^{2 x} x^2\right )\right )-2 \int \frac {1}{x \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+\left (4 \sqrt {\frac {3}{7}}\right ) \int \frac {1}{\left (-3+\sqrt {21}-2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+\left (4 \sqrt {\frac {3}{7}}\right ) \int \frac {1}{\left (3+\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx-\frac {12 \int \frac {1}{\left (-3+\sqrt {17}-2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx}{\sqrt {17}}-\frac {12 \int \frac {1}{\left (3+\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx}{\sqrt {17}}+\frac {1}{17} \left (4 \left (17-3 \sqrt {17}\right )\right ) \int \frac {1}{\left (3-\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx+\frac {1}{17} \left (4 \left (17+3 \sqrt {17}\right )\right ) \int \frac {1}{\left (3+\sqrt {17}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx-\frac {1}{7} \left (4 \left (7-\sqrt {21}\right )\right ) \int \frac {1}{\left (3-\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx-\frac {1}{7} \left (4 \left (7+\sqrt {21}\right )\right ) \int \frac {1}{\left (3+\sqrt {21}+2 x\right ) \log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 43, normalized size = 1.39 \begin {gather*} 2 \left (-\frac {1}{2} \log \left (\log \left (e^{2 x} x^2\right )\right )-\log \left (\log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12 + 24*x - 12*x^2 - 12*x^3 - 2*x^4)*Log[E^(2*x)*x^2] + (-12 + 18*x + 22*x^2 - 20*x^3 - 14*x^4 -
2*x^5)*Log[(3*x - 3*x^2 - x^3)/(-2 + 3*x + x^2)])/((6*x - 15*x^2 + 4*x^3 + 6*x^4 + x^5)*Log[E^(2*x)*x^2]*Log[(
3*x - 3*x^2 - x^3)/(-2 + 3*x + x^2)]),x]

[Out]

2*(-1/2*Log[Log[E^(2*x)*x^2]] - Log[Log[-((x*(-3 + 3*x + x^2))/(-2 + 3*x + x^2))]])

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 41, normalized size = 1.32 \begin {gather*} -\log \left (\log \left (x^{2} e^{\left (2 \, x\right )}\right )\right ) - 2 \, \log \left (\log \left (-\frac {x^{3} + 3 \, x^{2} - 3 \, x}{x^{2} + 3 \, x - 2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4-20*x^3+22*x^2+18*x-12)*log((-x^3-3*
x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, alg
orithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.45, size = 36, normalized size = 1.16 \begin {gather*} -\log \left (x + \log \relax (x)\right ) - 2 \, \log \left (-\log \left (x^{2} + 3 \, x - 2\right ) + \log \left (-x^{2} - 3 \, x + 3\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4-20*x^3+22*x^2+18*x-12)*log((-x^3-3*
x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, alg
orithm="giac")

[Out]

-log(x + log(x)) - 2*log(-log(x^2 + 3*x - 2) + log(-x^2 - 3*x + 3) + log(x))

________________________________________________________________________________________

maple [A]  time = 0.67, size = 43, normalized size = 1.39

method	result	size
default	\(-\ln \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )-2 \ln \left (\ln \left (\frac {-x^{3}-3 x^{2}+3 x}{x^{2}+3 x -2}\right )\right )\)	\(43\)
risch	\(-2 \ln \left (\ln \left (x^{2}+3 x -2\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x^{2}+3 x -3\right )\right ) \mathrm {csgn}\left (\frac {i}{x^{2}+3 x -2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )-\pi \,\mathrm {csgn}\left (i \left (x^{2}+3 x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}+3 x -2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right ) \mathrm {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )^{3}+2 i \ln \relax (x )+2 i \ln \left (x^{2}+3 x -3\right )-2 \pi \right )}{2}\right )-\ln \left (\ln \left ({\mathrm e}^{x}\right )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )-2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3}+4 i \ln \relax (x )\right )}{4}\right )\)	\(585\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4-12*x^3-12*x^2+24*x-12)*ln(exp(x)^2*x^2)+(-2*x^5-14*x^4-20*x^3+22*x^2+18*x-12)*ln((-x^3-3*x^2+3*x)
/(x^2+3*x-2)))/(x^5+6*x^4+4*x^3-15*x^2+6*x)/ln((-x^3-3*x^2+3*x)/(x^2+3*x-2))/ln(exp(x)^2*x^2),x,method=_RETURN
VERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 36, normalized size = 1.16 \begin {gather*} -\log \left (x + \log \relax (x)\right ) - 2 \, \log \left (-\log \left (x^{2} + 3 \, x - 2\right ) + \log \left (-x^{2} - 3 \, x + 3\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4-20*x^3+22*x^2+18*x-12)*log((-x^3-3*
x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, alg
orithm="maxima")

[Out]

-log(x + log(x)) - 2*log(-log(x^2 + 3*x - 2) + log(-x^2 - 3*x + 3) + log(x))

________________________________________________________________________________________

mupad [B]  time = 3.21, size = 40, normalized size = 1.29 \begin {gather*} -\ln \left (2\,x+\ln \left (x^2\right )\right )-2\,\ln \left (\ln \left (-\frac {x^3+3\,x^2-3\,x}{x^2+3\,x-2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2*exp(2*x))*(12*x^2 - 24*x + 12*x^3 + 2*x^4 + 12) + log(-(3*x^2 - 3*x + x^3)/(3*x + x^2 - 2))*(20*
x^3 - 22*x^2 - 18*x + 14*x^4 + 2*x^5 + 12))/(log(-(3*x^2 - 3*x + x^3)/(3*x + x^2 - 2))*log(x^2*exp(2*x))*(6*x
- 15*x^2 + 4*x^3 + 6*x^4 + x^5)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.96, size = 37, normalized size = 1.19 \begin {gather*} - \log {\left (\log {\left (x^{2} e^{2 x} \right )} \right )} - 2 \log {\left (\log {\left (\frac {- x^{3} - 3 x^{2} + 3 x}{x^{2} + 3 x - 2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4-12*x**3-12*x**2+24*x-12)*ln(exp(x)**2*x**2)+(-2*x**5-14*x**4-20*x**3+22*x**2+18*x-12)*ln((
-x**3-3*x**2+3*x)/(x**2+3*x-2)))/(x**5+6*x**4+4*x**3-15*x**2+6*x)/ln((-x**3-3*x**2+3*x)/(x**2+3*x-2))/ln(exp(x
)**2*x**2),x)

[Out]

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

________________________________________________________________________________________