3.9.77 \(\int \frac {1}{x-\sqrt {1+x^2}} \, dx\) [877]

Optimal. Leaf size=28 \[ -\frac {x^2}{2}-\frac {1}{2} x \sqrt {1+x^2}-\frac {1}{2} \sinh ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2131, 30, 201, 221} \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \sqrt {x^2+1} x-\frac {1}{2} \sinh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x - Sqrt[1 + x^2])^(-1),x]

[Out]

-1/2*x^2 - (x*Sqrt[1 + x^2])/2 - ArcSinh[x]/2

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

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 2131

Int[(u_.)/((d_.)*(x_)^(n_.) + (c_.)*Sqrt[(a_.) + (b_.)*(x_)^(p_.)]), x_Symbol] :> Dist[-b/(a*d), Int[u*x^n, x]
, x] + Dist[1/(a*c), Int[u*Sqrt[a + b*x^(2*n)], x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^
2 - d^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.18 \begin {gather*} \frac {1}{2} \left (-x \left (x+\sqrt {1+x^2}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - Sqrt[1 + x^2])^(-1),x]

[Out]

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

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Maple [A]
time = 0.51, size = 21, normalized size = 0.75

method	result	size
default	\(-\frac {x^{2}}{2}-\frac {\arcsinh \left (x \right )}{2}-\frac {x \sqrt {x^{2}+1}}{2}\)	\(21\)
trager	\(-\frac {x^{2}}{2}-\frac {x \sqrt {x^{2}+1}}{2}-\frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-(x^2+1)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x - sqrt(x^2 + 1)), x)

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Fricas [A]
time = 0.35, size = 30, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
time = 0.18, size = 58, normalized size = 2.07 \begin {gather*} - \frac {x \operatorname {asinh}{\left (x \right )}}{2 x - 2 \sqrt {x^{2} + 1}} + \frac {x}{2 x - 2 \sqrt {x^{2} + 1}} + \frac {\sqrt {x^{2} + 1} \operatorname {asinh}{\left (x \right )}}{2 x - 2 \sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.45, size = 30, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 20, normalized size = 0.71 \begin {gather*} -\frac {\mathrm {asinh}\left (x\right )}{2}-\frac {x\,\sqrt {x^2+1}}{2}-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - (x^2 + 1)^(1/2)),x)

[Out]

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

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