### 3.345 $$\int \frac{\sqrt{-1+9 x^2}}{x^2} \, dx$$

Optimal. Leaf size=34 $3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x}$

[Out]

-(Sqrt[-1 + 9*x^2]/x) + 3*ArcTanh[(3*x)/Sqrt[-1 + 9*x^2]]

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Rubi [A]  time = 0.007611, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {277, 217, 206} $3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{9 x^2-1}}\right )-\frac{\sqrt{9 x^2-1}}{x}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[-1 + 9*x^2]/x^2,x]

[Out]

-(Sqrt[-1 + 9*x^2]/x) + 3*ArcTanh[(3*x)/Sqrt[-1 + 9*x^2]]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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{\sqrt{-1+9 x^2}}{x^2} \, dx &=-\frac{\sqrt{-1+9 x^2}}{x}+9 \int \frac{1}{\sqrt{-1+9 x^2}} \, dx\\ &=-\frac{\sqrt{-1+9 x^2}}{x}+9 \operatorname{Subst}\left (\int \frac{1}{1-9 x^2} \, dx,x,\frac{x}{\sqrt{-1+9 x^2}}\right )\\ &=-\frac{\sqrt{-1+9 x^2}}{x}+3 \tanh ^{-1}\left (\frac{3 x}{\sqrt{-1+9 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0152595, size = 35, normalized size = 1.03 $\sqrt{9 x^2-1} \left (-\frac{3 \sin ^{-1}(3 x)}{\sqrt{1-9 x^2}}-\frac{1}{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[-1 + 9*x^2]/x^2,x]

[Out]

Sqrt[-1 + 9*x^2]*(-x^(-1) - (3*ArcSin[3*x])/Sqrt[1 - 9*x^2])

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Maple [A]  time = 0.005, size = 47, normalized size = 1.4 \begin{align*}{\frac{1}{x} \left ( 9\,{x}^{2}-1 \right ) ^{{\frac{3}{2}}}}-9\,x\sqrt{9\,{x}^{2}-1}+\ln \left ( x\sqrt{9}+\sqrt{9\,{x}^{2}-1} \right ) \sqrt{9} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2-1)^(1/2)/x^2,x)

[Out]

1/x*(9*x^2-1)^(3/2)-9*x*(9*x^2-1)^(1/2)+ln(x*9^(1/2)+(9*x^2-1)^(1/2))*9^(1/2)

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Maxima [A]  time = 1.4237, size = 45, normalized size = 1.32 \begin{align*} -\frac{\sqrt{9 \, x^{2} - 1}}{x} + 3 \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} - 1}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-1)^(1/2)/x^2,x, algorithm="maxima")

[Out]

-sqrt(9*x^2 - 1)/x + 3*log(18*x + 6*sqrt(9*x^2 - 1))

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Fricas [A]  time = 1.89075, size = 84, normalized size = 2.47 \begin{align*} -\frac{3 \, x \log \left (-3 \, x + \sqrt{9 \, x^{2} - 1}\right ) + 3 \, x + \sqrt{9 \, x^{2} - 1}}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-1)^(1/2)/x^2,x, algorithm="fricas")

[Out]

-(3*x*log(-3*x + sqrt(9*x^2 - 1)) + 3*x + sqrt(9*x^2 - 1))/x

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Sympy [A]  time = 0.224803, size = 17, normalized size = 0.5 \begin{align*} 3 \operatorname{acosh}{\left (3 x \right )} - \frac{\sqrt{9 x^{2} - 1}}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2-1)**(1/2)/x**2,x)

[Out]

3*acosh(3*x) - sqrt(9*x**2 - 1)/x

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Giac [A]  time = 1.073, size = 59, normalized size = 1.74 \begin{align*} -\frac{6}{{\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2} + 1} - \frac{3}{2} \, \log \left ({\left (3 \, x - \sqrt{9 \, x^{2} - 1}\right )}^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-1)^(1/2)/x^2,x, algorithm="giac")

[Out]

-6/((3*x - sqrt(9*x^2 - 1))^2 + 1) - 3/2*log((3*x - sqrt(9*x^2 - 1))^2)