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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \sinh ^{-1}(x)-\frac {2 x}{\sqrt {x^2+1}} \]

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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fricas [B]  time = 0.63, size = 44, normalized size = 2.93 \[ -\frac {2 \, x^{2} + {\left (x^{2} + 1\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) + 2 \, \sqrt {x^{2} + 1} x + 2}{x^{2} + 1} \]

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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giac [A]  time = 0.57, size = 25, normalized size = 1.67 \[ -\frac {2 \, x}{\sqrt {x^{2} + 1}} - \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

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))

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maple [A]  time = 0.01, size = 14, normalized size = 0.93 \[ -\frac {2 x}{\sqrt {x^{2}+1}}+\arcsinh \relax (x ) \]

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 = 2.78, size = 13, normalized size = 0.87 \[ -\frac {2 \, x}{\sqrt {x^{2} + 1}} + \operatorname {arsinh}\relax (x) \]

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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mupad [B]  time = 0.04, size = 27, normalized size = 1.80 \[ \frac {\mathrm {asinh}\relax (x)+x^2\,\mathrm {asinh}\relax (x)-2\,x\,\sqrt {x^2+1}}{x^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 1)/(x^2 + 1)^(3/2),x)

[Out]

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

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sympy [B]  time = 4.68, size = 31, normalized size = 2.07 \[ \frac {x^{2} \operatorname {asinh}{\relax (x )}}{x^{2} + 1} - \frac {2 x}{\sqrt {x^{2} + 1}} + \frac {\operatorname {asinh}{\relax (x )}}{x^{2} + 1} \]

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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