3.169 \(\int \frac{5-4 x}{\left (-2-4 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac{7 \tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{20 \sqrt{10}} \]

[Out]

-(18 - 7*x)/(20*(2 + 4*x - 3*x^2)) - (7*ArcTanh[(2 - 3*x)/Sqrt[10]])/(20*Sqrt[10
])

_______________________________________________________________________________________

Rubi [A]  time = 0.0507928, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac{7 \tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - 4*x)/(-2 - 4*x + 3*x^2)^2,x]

[Out]

-(18 - 7*x)/(20*(2 + 4*x - 3*x^2)) - (7*ArcTanh[(2 - 3*x)/Sqrt[10]])/(20*Sqrt[10
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.29662, size = 37, normalized size = 0.86 \[ - \frac{- 14 x + 36}{40 \left (- 3 x^{2} + 4 x + 2\right )} + \frac{7 \sqrt{10} \operatorname{atanh}{\left (\sqrt{10} \left (\frac{3 x}{10} - \frac{1}{5}\right ) \right )}}{200} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-4*x)/(3*x**2-4*x-2)**2,x)

[Out]

-(-14*x + 36)/(40*(-3*x**2 + 4*x + 2)) + 7*sqrt(10)*atanh(sqrt(10)*(3*x/10 - 1/5
))/200

_______________________________________________________________________________________

Mathematica [A]  time = 0.0604995, size = 62, normalized size = 1.44 \[ \frac{18-7 x}{20 \left (3 x^2-4 x-2\right )}-\frac{7 \log \left (-3 x+\sqrt{10}+2\right )}{40 \sqrt{10}}+\frac{7 \log \left (3 x+\sqrt{10}-2\right )}{40 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - 4*x)/(-2 - 4*x + 3*x^2)^2,x]

[Out]

(18 - 7*x)/(20*(-2 - 4*x + 3*x^2)) - (7*Log[2 + Sqrt[10] - 3*x])/(40*Sqrt[10]) +
(7*Log[-2 + Sqrt[10] + 3*x])/(40*Sqrt[10])

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 37, normalized size = 0.9 \[ -{\frac{14\,x-36}{120\,{x}^{2}-160\,x-80}}+{\frac{7\,\sqrt{10}}{200}{\it Artanh} \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{10}}{20}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-4*x)/(3*x^2-4*x-2)^2,x)

[Out]

-1/40*(14*x-36)/(3*x^2-4*x-2)+7/200*10^(1/2)*arctanh(1/20*(6*x-4)*10^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.59646, size = 63, normalized size = 1.47 \[ -\frac{7}{400} \, \sqrt{10} \log \left (\frac{3 \, x - \sqrt{10} - 2}{3 \, x + \sqrt{10} - 2}\right ) - \frac{7 \, x - 18}{20 \,{\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x - 5)/(3*x^2 - 4*x - 2)^2,x, algorithm="maxima")

[Out]

-7/400*sqrt(10)*log((3*x - sqrt(10) - 2)/(3*x + sqrt(10) - 2)) - 1/20*(7*x - 18)
/(3*x^2 - 4*x - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.197833, size = 99, normalized size = 2.3 \[ \frac{\sqrt{10}{\left (7 \,{\left (3 \, x^{2} - 4 \, x - 2\right )} \log \left (\frac{\sqrt{10}{\left (9 \, x^{2} - 12 \, x + 14\right )} + 60 \, x - 40}{3 \, x^{2} - 4 \, x - 2}\right ) - 2 \, \sqrt{10}{\left (7 \, x - 18\right )}\right )}}{400 \,{\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x - 5)/(3*x^2 - 4*x - 2)^2,x, algorithm="fricas")

[Out]

1/400*sqrt(10)*(7*(3*x^2 - 4*x - 2)*log((sqrt(10)*(9*x^2 - 12*x + 14) + 60*x - 4
0)/(3*x^2 - 4*x - 2)) - 2*sqrt(10)*(7*x - 18))/(3*x^2 - 4*x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.149857, size = 58, normalized size = 1.35 \[ - \frac{7 x - 18}{60 x^{2} - 80 x - 40} + \frac{7 \sqrt{10} \log{\left (x - \frac{2}{3} + \frac{\sqrt{10}}{3} \right )}}{400} - \frac{7 \sqrt{10} \log{\left (x - \frac{\sqrt{10}}{3} - \frac{2}{3} \right )}}{400} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-4*x)/(3*x**2-4*x-2)**2,x)

[Out]

-(7*x - 18)/(60*x**2 - 80*x - 40) + 7*sqrt(10)*log(x - 2/3 + sqrt(10)/3)/400 - 7
*sqrt(10)*log(x - sqrt(10)/3 - 2/3)/400

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221801, size = 69, normalized size = 1.6 \[ -\frac{7}{400} \, \sqrt{10}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}}\right ) - \frac{7 \, x - 18}{20 \,{\left (3 \, x^{2} - 4 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x - 5)/(3*x^2 - 4*x - 2)^2,x, algorithm="giac")

[Out]

-7/400*sqrt(10)*ln(abs(6*x - 2*sqrt(10) - 4)/abs(6*x + 2*sqrt(10) - 4)) - 1/20*(
7*x - 18)/(3*x^2 - 4*x - 2)