3.108 \(\int \frac{-1+x^2}{(1+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=15 \[ \sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}} \]

[Out]

(-2*x)/Sqrt[1 + x^2] + ArcSinh[x]

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Rubi [A]  time = 0.0040774, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {385, 215} \[ \sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/(1 + x^2)^(3/2),x]

[Out]

(-2*x)/Sqrt[1 + x^2] + ArcSinh[x]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0119162, size = 15, normalized size = 1. \[ \sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[1 + x^2] + ArcSinh[x]

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ( x \right ) -2\,{\frac{x}{\sqrt{{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(x)-2*x/(x^2+1)^(1/2)

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Maxima [A]  time = 1.45842, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, x}{\sqrt{x^{2} + 1}} + \operatorname{arsinh}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/sqrt(x^2 + 1) + arcsinh(x)

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Fricas [B]  time = 1.51439, size = 108, normalized size = 7.2 \begin{align*} -\frac{2 \, x^{2} +{\left (x^{2} + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) + 2 \, \sqrt{x^{2} + 1} x + 2}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.45313, size = 31, normalized size = 2.07 \begin{align*} \frac{x^{2} \operatorname{asinh}{\left (x \right )}}{x^{2} + 1} - \frac{2 x}{\sqrt{x^{2} + 1}} + \frac{\operatorname{asinh}{\left (x \right )}}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*asinh(x)/(x**2 + 1) - 2*x/sqrt(x**2 + 1) + asinh(x)/(x**2 + 1)

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Giac [A]  time = 1.08485, size = 34, normalized size = 2.27 \begin{align*} -\frac{2 \, x}{\sqrt{x^{2} + 1}} - \log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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