3.99 \(\int \frac {-5+2 x^2}{6-5 x^2+x^4} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1166, 207} \[ -\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(-5 + 2*x^2)/(6 - 5*x^2 + x^4),x]

[Out]

-(ArcTanh[x/Sqrt[2]]/Sqrt[2]) - ArcTanh[x/Sqrt[3]]/Sqrt[3]

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])

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]

Rubi steps

\begin {align*} \int \frac {-5+2 x^2}{6-5 x^2+x^4} \, dx &=\int \frac {1}{-3+x^2} \, dx+\int \frac {1}{-2+x^2} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 69, normalized size = 2.23 \[ \frac {1}{12} \left (3 \sqrt {2} \log \left (\sqrt {2}-x\right )+2 \sqrt {3} \log \left (\sqrt {3}-x\right )-3 \sqrt {2} \log \left (x+\sqrt {2}\right )-2 \sqrt {3} \log \left (x+\sqrt {3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 2*x^2)/(6 - 5*x^2 + x^4),x]

[Out]

(3*Sqrt[2]*Log[Sqrt[2] - x] + 2*Sqrt[3]*Log[Sqrt[3] - x] - 3*Sqrt[2]*Log[Sqrt[2] + x] - 2*Sqrt[3]*Log[Sqrt[3]
+ x])/12

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IntegrateAlgebraic [A]  time = 0.04, size = 31, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-5 + 2*x^2)/(6 - 5*x^2 + x^4),x]

[Out]

-(ArcTanh[x/Sqrt[2]]/Sqrt[2]) - ArcTanh[x/Sqrt[3]]/Sqrt[3]

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fricas [B]  time = 0.88, size = 51, normalized size = 1.65 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{2} - 2 \, \sqrt {2} x + 2}{x^{2} - 2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-5)/(x^4-5*x^2+6),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log((x^2 - 2*sqrt(2)*x + 2)/(x^2 - 2)) + 1/6*sqrt(3)*log((x^2 - 2*sqrt(3)*x + 3)/(x^2 - 3))

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giac [B]  time = 0.99, size = 59, normalized size = 1.90 \[ \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-5)/(x^4-5*x^2+6),x, algorithm="giac")

[Out]

1/6*sqrt(3)*log(abs(2*x - 2*sqrt(3))/abs(2*x + 2*sqrt(3))) + 1/4*sqrt(2)*log(abs(2*x - 2*sqrt(2))/abs(2*x + 2*
sqrt(2)))

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maple [A]  time = 0.04, size = 26, normalized size = 0.84

method	result	size
default	\(-\frac {\arctanh \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\arctanh \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(26\)
risch	\(\frac {\sqrt {3}\, \ln \left (x -\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (x +\sqrt {3}\right )}{6}+\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right )}{4}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-5)/(x^4-5*x^2+6),x,method=_RETURNVERBOSE)

[Out]

-1/2*arctanh(1/2*x*2^(1/2))*2^(1/2)-1/3*arctanh(1/3*x*3^(1/2))*3^(1/2)

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maxima [A]  time = 0.98, size = 43, normalized size = 1.39 \[ \frac {1}{6} \, \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-5)/(x^4-5*x^2+6),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*log((x - sqrt(3))/(x + sqrt(3))) + 1/4*sqrt(2)*log((x - sqrt(2))/(x + sqrt(2)))

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mupad [B]  time = 0.08, size = 25, normalized size = 0.81 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - 5)/(x^4 - 5*x^2 + 6),x)

[Out]

- (2^(1/2)*atanh((2^(1/2)*x)/2))/2 - (3^(1/2)*atanh((3^(1/2)*x)/3))/3

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sympy [A]  time = 0.58, size = 60, normalized size = 1.94 \[ \frac {\sqrt {2} \log {\left (x - \sqrt {2} \right )}}{4} - \frac {\sqrt {2} \log {\left (x + \sqrt {2} \right )}}{4} + \frac {\sqrt {3} \log {\left (x - \sqrt {3} \right )}}{6} - \frac {\sqrt {3} \log {\left (x + \sqrt {3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-5)/(x**4-5*x**2+6),x)

[Out]

sqrt(2)*log(x - sqrt(2))/4 - sqrt(2)*log(x + sqrt(2))/4 + sqrt(3)*log(x - sqrt(3))/6 - sqrt(3)*log(x + sqrt(3)
)/6

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