3.103 \(\int \frac{2+3 x^2}{5-8 x^2+3 x^4} \, dx\)

Optimal. Leaf size=28 \[ \frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right ) \]

[Out]

(5*ArcTanh[x])/2 - (7*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*x])/2

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Rubi [A]  time = 0.0313043, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4),x]

[Out]

(5*ArcTanh[x])/2 - (7*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*x])/2

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Rubi in Sympy [A]  time = 8.6189, size = 24, normalized size = 0.86 \[ \frac{5 \operatorname{atanh}{\left (x \right )}}{2} - \frac{7 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} x}{5} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/(3*x**4-8*x**2+5),x)

[Out]

5*atanh(x)/2 - 7*sqrt(15)*atanh(sqrt(15)*x/5)/10

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Mathematica [A]  time = 0.0304294, size = 53, normalized size = 1.89 \[ \frac{1}{20} \left (7 \sqrt{15} \log \left (\sqrt{15}-3 x\right )-25 \log (1-x)+25 \log (x+1)-7 \sqrt{15} \log \left (3 x+\sqrt{15}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4),x]

[Out]

(7*Sqrt[15]*Log[Sqrt[15] - 3*x] - 25*Log[1 - x] + 25*Log[1 + x] - 7*Sqrt[15]*Log
[Sqrt[15] + 3*x])/20

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Maple [A]  time = 0.01, size = 26, normalized size = 0.9 \[ -{\frac{5\,\ln \left ( -1+x \right ) }{4}}-{\frac{7\,\sqrt{15}}{10}{\it Artanh} \left ({\frac{x\sqrt{15}}{5}} \right ) }+{\frac{5\,\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/(3*x^4-8*x^2+5),x)

[Out]

-5/4*ln(-1+x)-7/10*arctanh(1/5*x*15^(1/2))*15^(1/2)+5/4*ln(1+x)

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Maxima [A]  time = 0.847103, size = 51, normalized size = 1.82 \[ \frac{7}{20} \, \sqrt{15} \log \left (\frac{3 \, x - \sqrt{15}}{3 \, x + \sqrt{15}}\right ) + \frac{5}{4} \, \log \left (x + 1\right ) - \frac{5}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="maxima")

[Out]

7/20*sqrt(15)*log((3*x - sqrt(15))/(3*x + sqrt(15))) + 5/4*log(x + 1) - 5/4*log(
x - 1)

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Fricas [A]  time = 0.286182, size = 78, normalized size = 2.79 \[ \frac{1}{20} \, \sqrt{5}{\left (5 \, \sqrt{5} \log \left (x + 1\right ) - 5 \, \sqrt{5} \log \left (x - 1\right ) + 7 \, \sqrt{3} \log \left (\frac{\sqrt{5}{\left (3 \, x^{2} + 5\right )} - 10 \, \sqrt{3} x}{3 \, x^{2} - 5}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="fricas")

[Out]

1/20*sqrt(5)*(5*sqrt(5)*log(x + 1) - 5*sqrt(5)*log(x - 1) + 7*sqrt(3)*log((sqrt(
5)*(3*x^2 + 5) - 10*sqrt(3)*x)/(3*x^2 - 5)))

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Sympy [A]  time = 1.85561, size = 53, normalized size = 1.89 \[ - \frac{5 \log{\left (x - 1 \right )}}{4} + \frac{5 \log{\left (x + 1 \right )}}{4} + \frac{7 \sqrt{15} \log{\left (x - \frac{\sqrt{15}}{3} \right )}}{20} - \frac{7 \sqrt{15} \log{\left (x + \frac{\sqrt{15}}{3} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/(3*x**4-8*x**2+5),x)

[Out]

-5*log(x - 1)/4 + 5*log(x + 1)/4 + 7*sqrt(15)*log(x - sqrt(15)/3)/20 - 7*sqrt(15
)*log(x + sqrt(15)/3)/20

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GIAC/XCAS [A]  time = 0.269819, size = 59, normalized size = 2.11 \[ \frac{7}{20} \, \sqrt{15}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{15} \right |}}{{\left | 6 \, x + 2 \, \sqrt{15} \right |}}\right ) + \frac{5}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{5}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="giac")

[Out]

7/20*sqrt(15)*ln(abs(6*x - 2*sqrt(15))/abs(6*x + 2*sqrt(15))) + 5/4*ln(abs(x + 1
)) - 5/4*ln(abs(x - 1))