3.42.89 \(\int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx\)

Optimal. Leaf size=26 \[ 81-\frac {x}{1-\frac {2 (2 x-\log (2))}{3+81 x}} \]

________________________________________________________________________________________

Rubi [A]  time = 2.39, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1984, 27, 683} \begin {gather*} -\frac {81 x}{77}-\frac {3 \left (12+27 \log ^2(4)-154 \log (2)+162 \log (4)\right )}{5929 (77 x+3+\log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 486*x - 6237*x^2 + (-6 - 324*x)*Log[2])/(9 + 462*x + 5929*x^2 + (12 + 308*x)*Log[2] + 4*Log[2]^2),x]

[Out]

(-81*x)/77 - (3*(12 - 154*Log[2] + 162*Log[4] + 27*Log[4]^2))/(5929*(3 + 77*x + Log[4]))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6237 x^2-3 (3+2 \log (2))-162 x (3+\log (4))}{5929 x^2+154 x (3+\log (4))+(3+\log (4))^2} \, dx\\ &=\int \frac {-6237 x^2-3 (3+2 \log (2))-162 x (3+\log (4))}{(3+77 x+\log (4))^2} \, dx\\ &=\int \left (-\frac {81}{77}+\frac {3 \left (12-154 \log (2)+162 \log (4)+27 \log ^2(4)\right )}{77 (3+77 x+\log (4))^2}\right ) \, dx\\ &=-\frac {81 x}{77}-\frac {3 \left (12-154 \log (2)+162 \log (4)+27 \log ^2(4)\right )}{5929 (3+77 x+\log (4))}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 37, normalized size = 1.42 \begin {gather*} -\frac {3 \left (255+160083 x^2+247 \log (4)+54 \log ^2(4)+4158 x (3+\log (4))\right )}{5929 (3+77 x+\log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 486*x - 6237*x^2 + (-6 - 324*x)*Log[2])/(9 + 462*x + 5929*x^2 + (12 + 308*x)*Log[2] + 4*Log[2]
^2),x]

[Out]

(-3*(255 + 160083*x^2 + 247*Log[4] + 54*Log[4]^2 + 4158*x*(3 + Log[4])))/(5929*(3 + 77*x + Log[4]))

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 38, normalized size = 1.46 \begin {gather*} -\frac {3 \, {\left (160083 \, x^{2} + 2 \, {\left (2079 \, x + 85\right )} \log \relax (2) + 108 \, \log \relax (2)^{2} + 6237 \, x + 12\right )}}{5929 \, {\left (77 \, x + 2 \, \log \relax (2) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log(2)+5929*x^2+462*x+9),x, algorithm="f
ricas")

[Out]

-3/5929*(160083*x^2 + 2*(2079*x + 85)*log(2) + 108*log(2)^2 + 6237*x + 12)/(77*x + 2*log(2) + 3)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 29, normalized size = 1.12 \begin {gather*} -\frac {81}{77} \, x - \frac {6 \, {\left (54 \, \log \relax (2)^{2} + 85 \, \log \relax (2) + 6\right )}}{5929 \, {\left (77 \, x + 2 \, \log \relax (2) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log(2)+5929*x^2+462*x+9),x, algorithm="g
iac")

[Out]

-81/77*x - 6/5929*(54*log(2)^2 + 85*log(2) + 6)/(77*x + 2*log(2) + 3)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 24, normalized size = 0.92

method	result	size
norman	\(\frac {-81 x^{2}+\frac {6 \ln \relax (2)}{77}+\frac {9}{77}}{2 \ln \relax (2)+77 x +3}\)	\(24\)
gosper	\(\frac {-81 x^{2}+\frac {6 \ln \relax (2)}{77}+\frac {9}{77}}{2 \ln \relax (2)+77 x +3}\)	\(25\)
default	\(-\frac {81 x}{77}-\frac {3 \left (\frac {108 \ln \relax (2)^{2}}{5929}+\frac {170 \ln \relax (2)}{5929}+\frac {12}{5929}\right )}{2 \ln \relax (2)+77 x +3}\)	\(30\)
risch	\(-\frac {81 x}{77}-\frac {162 \ln \relax (2)^{2}}{5929 \left (\ln \relax (2)+\frac {77 x}{2}+\frac {3}{2}\right )}-\frac {255 \ln \relax (2)}{5929 \left (\ln \relax (2)+\frac {77 x}{2}+\frac {3}{2}\right )}-\frac {18}{5929 \left (\ln \relax (2)+\frac {77 x}{2}+\frac {3}{2}\right )}\)	\(44\)
meijerg	\(-\frac {9 x}{77 \left (\frac {2 \ln \relax (2)}{77}+\frac {3}{77}\right ) \left (1+\frac {77 x}{2 \ln \relax (2)+3}\right ) \left (2 \ln \relax (2)+3\right )}-\frac {81 \left (2 \ln \relax (2)+3\right )^{2} \left (\frac {77 x \left (\frac {231 x}{2 \ln \relax (2)+3}+6\right )}{3 \left (2 \ln \relax (2)+3\right ) \left (1+\frac {77 x}{2 \ln \relax (2)+3}\right )}-2 \ln \left (1+\frac {77 x}{2 \ln \relax (2)+3}\right )\right )}{456533 \left (\frac {2 \ln \relax (2)}{77}+\frac {3}{77}\right )}+\frac {\left (-324 \ln \relax (2)-486\right ) \left (-\frac {77 x}{\left (1+\frac {77 x}{2 \ln \relax (2)+3}\right ) \left (2 \ln \relax (2)+3\right )}+\ln \left (1+\frac {77 x}{2 \ln \relax (2)+3}\right )\right )}{5929}-\frac {6 \ln \relax (2) x}{77 \left (\frac {2 \ln \relax (2)}{77}+\frac {3}{77}\right ) \left (1+\frac {77 x}{2 \ln \relax (2)+3}\right ) \left (2 \ln \relax (2)+3\right )}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-324*x-6)*ln(2)-6237*x^2-486*x-9)/(4*ln(2)^2+(308*x+12)*ln(2)+5929*x^2+462*x+9),x,method=_RETURNVERBOSE)

[Out]

(-81*x^2+6/77*ln(2)+9/77)/(2*ln(2)+77*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 29, normalized size = 1.12 \begin {gather*} -\frac {81}{77} \, x - \frac {6 \, {\left (54 \, \log \relax (2)^{2} + 85 \, \log \relax (2) + 6\right )}}{5929 \, {\left (77 \, x + 2 \, \log \relax (2) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log(2)+5929*x^2+462*x+9),x, algorithm="m
axima")

[Out]

-81/77*x - 6/5929*(54*log(2)^2 + 85*log(2) + 6)/(77*x + 2*log(2) + 3)

________________________________________________________________________________________

mupad [B]  time = 2.97, size = 27, normalized size = 1.04 \begin {gather*} -\frac {81\,x}{77}-\frac {\frac {510\,\ln \relax (2)}{5929}+\frac {324\,{\ln \relax (2)}^2}{5929}+\frac {36}{5929}}{77\,x+\ln \relax (4)+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(486*x + log(2)*(324*x + 6) + 6237*x^2 + 9)/(462*x + log(2)*(308*x + 12) + 4*log(2)^2 + 5929*x^2 + 9),x)

[Out]

- (81*x)/77 - ((510*log(2))/5929 + (324*log(2)^2)/5929 + 36/5929)/(77*x + log(4) + 3)

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 29, normalized size = 1.12 \begin {gather*} - \frac {81 x}{77} - \frac {36 + 324 \log {\relax (2 )}^{2} + 510 \log {\relax (2 )}}{456533 x + 11858 \log {\relax (2 )} + 17787} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x-6)*ln(2)-6237*x**2-486*x-9)/(4*ln(2)**2+(308*x+12)*ln(2)+5929*x**2+462*x+9),x)

[Out]

-81*x/77 - (36 + 324*log(2)**2 + 510*log(2))/(456533*x + 11858*log(2) + 17787)

________________________________________________________________________________________