3.311 \(\int \frac{-1+x^2}{x \sqrt{1+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=21 \[ \tanh ^{-1}\left (\frac{x^2+1}{\sqrt{x^4+3 x^2+1}}\right ) \]

[Out]

ArcTanh[(1 + x^2)/Sqrt[1 + 3*x^2 + x^4]]

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Rubi [A]  time = 0.0291543, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1251, 838, 206} \[ \tanh ^{-1}\left (\frac{x^2+1}{\sqrt{x^4+3 x^2+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/(x*Sqrt[1 + 3*x^2 + x^4]),x]

[Out]

ArcTanh[(1 + x^2)/Sqrt[1 + 3*x^2 + x^4]]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 838

Int[((f_) + (g_.)*(x_))/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(4*f
*(a - d))/(b*d - a*e), Subst[Int[1/(4*(a - d) - x^2), x], x, (2*(a - d) + (b - e)*x)/Sqrt[a + b*x + c*x^2]], x
] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[4*c*(a - d) - (b - e)^2, 0] && EqQ[e*f*(b - e) - 2*g*(b*d - a*e),
0] && NeQ[b*d - a*e, 0]

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 0.0153564, size = 57, normalized size = 2.71 \[ \frac{1}{2} \left (\tanh ^{-1}\left (\frac{2 x^2+3}{2 \sqrt{x^4+3 x^2+1}}\right )+\tanh ^{-1}\left (\frac{3 x^2+2}{2 \sqrt{x^4+3 x^2+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)/(x*Sqrt[1 + 3*x^2 + x^4]),x]

[Out]

(ArcTanh[(3 + 2*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])] + ArcTanh[(2 + 3*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])])/2

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Maple [B]  time = 0.017, size = 46, normalized size = 2.2 \begin{align*}{\frac{1}{2}{\it Artanh} \left ({\frac{3\,{x}^{2}+2}{2}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+1}}}} \right ) }+{\frac{1}{2}\ln \left ({x}^{2}+{\frac{3}{2}}+\sqrt{{x}^{4}+3\,{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/x/(x^4+3*x^2+1)^(1/2),x)

[Out]

1/2*arctanh(1/2*(3*x^2+2)/(x^4+3*x^2+1)^(1/2))+1/2*ln(x^2+3/2+(x^4+3*x^2+1)^(1/2))

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Maxima [B]  time = 0.978983, size = 70, normalized size = 3.33 \begin{align*} \frac{1}{2} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 3 \, x^{2} + 1} + 3\right ) + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{x^{4} + 3 \, x^{2} + 1}}{x^{2}} + \frac{2}{x^{2}} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/x/(x^4+3*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.13552, size = 149, normalized size = 7.1 \begin{align*} -\frac{1}{2} \, \log \left (4 \, x^{4} + 11 \, x^{2} - \sqrt{x^{4} + 3 \, x^{2} + 1}{\left (4 \, x^{2} + 5\right )} + 5\right ) + \frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 3 \, x^{2} + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/x/(x^4+3*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/2*log(4*x^4 + 11*x^2 - sqrt(x^4 + 3*x^2 + 1)*(4*x^2 + 5) + 5) + 1/2*log(-x^2 + sqrt(x^4 + 3*x^2 + 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right ) \left (x + 1\right )}{x \sqrt{x^{4} + 3 x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)/x/(x**4+3*x**2+1)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)/(x*sqrt(x**4 + 3*x**2 + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} - 1}{\sqrt{x^{4} + 3 \, x^{2} + 1} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/x/(x^4+3*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/(sqrt(x^4 + 3*x^2 + 1)*x), x)