### 3.121 $$\int \frac{1}{\sqrt{-a^2+x^2}} \, dx$$

Optimal. Leaf size=16 $\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right )$

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

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Rubi [A]  time = 0.0024596, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {217, 206} $\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[-a^2 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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}{\sqrt{-a^2+x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-a^2+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{-a^2+x^2}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0027682, size = 46, normalized size = 2.88 $\frac{1}{2} \log \left (\frac{x}{\sqrt{x^2-a^2}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{x^2-a^2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/Sqrt[-a^2 + x^2],x]

[Out]

-Log[1 - x/Sqrt[-a^2 + x^2]]/2 + Log[1 + x/Sqrt[-a^2 + x^2]]/2

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} \ln \left ( x+\sqrt{-{a}^{2}+{x}^{2}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+(-a^2+x^2)^(1/2))

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Maxima [A]  time = 0.974328, size = 24, normalized size = 1.5 \begin{align*} \log \left (2 \, x + 2 \, \sqrt{-a^{2} + x^{2}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x + 2*sqrt(-a^2 + x^2))

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Fricas [A]  time = 1.88426, size = 39, normalized size = 2.44 \begin{align*} -\log \left (-x + \sqrt{-a^{2} + x^{2}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(-a^2 + x^2))

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Sympy [A]  time = 1.0204, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \operatorname{acosh}{\left (\frac{x}{a} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\- i \operatorname{asin}{\left (\frac{x}{a} \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((acosh(x/a), Abs(x**2)/Abs(a**2) > 1), (-I*asin(x/a), True))

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Giac [A]  time = 1.05721, size = 26, normalized size = 1.62 \begin{align*} -\log \left ({\left | -x + \sqrt{-a^{2} + x^{2}} \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x + sqrt(-a^2 + x^2)))