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

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 = 0.0126262, 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, Rules used = {1166, 207} \[ \frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right ) \]

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2+3 x^2}{5-8 x^2+3 x^4} \, dx &=-\left (\frac{15}{2} \int \frac{1}{-3+3 x^2} \, dx\right )+\frac{21}{2} \int \frac{1}{-5+3 x^2} \, dx\\ &=\frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right )\\ \end{align*}

Mathematica [A]  time = 0.0195733, 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 = 0.045, size = 26, normalized size = 0.9 \begin{align*}{\frac{5\,\ln \left ( 1+x \right ) }{4}}-{\frac{5\,\ln \left ( -1+x \right ) }{4}}-{\frac{7\,\sqrt{15}}{10}{\it Artanh} \left ({\frac{x\sqrt{15}}{5}} \right ) } \end{align*}

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

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Maxima [B]  time = 1.4477, size = 51, normalized size = 1.82 \begin{align*} \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 ) \end{align*}

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 [B]  time = 1.3812, size = 146, normalized size = 5.21 \begin{align*} \frac{7}{20} \, \sqrt{5} \sqrt{3} \log \left (-\frac{2 \, \sqrt{5} \sqrt{3} x - 3 \, x^{2} - 5}{3 \, x^{2} - 5}\right ) + \frac{5}{4} \, \log \left (x + 1\right ) - \frac{5}{4} \, \log \left (x - 1\right ) \end{align*}

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]

7/20*sqrt(5)*sqrt(3)*log(-(2*sqrt(5)*sqrt(3)*x - 3*x^2 - 5)/(3*x^2 - 5)) + 5/4*log(x + 1) - 5/4*log(x - 1)

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Sympy [B]  time = 0.489878, size = 53, normalized size = 1.89 \begin{align*} - \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} \end{align*}

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 [B]  time = 1.10018, size = 59, normalized size = 2.11 \begin{align*} \frac{7}{20} \, \sqrt{15} \log \left (\frac{{\left | 6 \, x - 2 \, \sqrt{15} \right |}}{{\left | 6 \, x + 2 \, \sqrt{15} \right |}}\right ) + \frac{5}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{5}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

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)*log(abs(6*x - 2*sqrt(15))/abs(6*x + 2*sqrt(15))) + 5/4*log(abs(x + 1)) - 5/4*log(abs(x - 1))