### 3.146 $$\int \frac{1}{\sqrt{-8+6 x+9 x^2}} \, dx$$

Optimal. Leaf size=25 $\frac{1}{3} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right )$

[Out]

ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]]/3

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Rubi [A]  time = 0.0058126, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {621, 206} $\frac{1}{3} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]]/3

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{-8+6 x+9 x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{6+18 x}{\sqrt{-8+6 x+9 x^2}}\right )\\ &=\frac{1}{3} \tanh ^{-1}\left (\frac{1+3 x}{\sqrt{-8+6 x+9 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0059543, size = 24, normalized size = 0.96 $\frac{1}{3} \log \left (\sqrt{9 x^2+6 x-8}+3 x+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

Log[1 + 3*x + Sqrt[-8 + 6*x + 9*x^2]]/3

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Maple [A]  time = 0.003, size = 30, normalized size = 1.2 \begin{align*}{\frac{\sqrt{9}}{9}\ln \left ({\frac{ \left ( 3+9\,x \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+6\,x-8} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2+6*x-8)^(1/2),x)

[Out]

1/9*ln(1/9*(3+9*x)*9^(1/2)+(9*x^2+6*x-8)^(1/2))*9^(1/2)

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Maxima [A]  time = 1.40184, size = 30, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 6 \, x - 8} + 6\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+6*x-8)^(1/2),x, algorithm="maxima")

[Out]

1/3*log(18*x + 6*sqrt(9*x^2 + 6*x - 8) + 6)

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Fricas [A]  time = 2.08571, size = 59, normalized size = 2.36 \begin{align*} -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+6*x-8)^(1/2),x, algorithm="fricas")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 + 6*x - 8) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{9 x^{2} + 6 x - 8}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2+6*x-8)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 + 6*x - 8), x)

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Giac [A]  time = 1.04942, size = 28, normalized size = 1.12 \begin{align*} -\frac{1}{3} \, \log \left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+6*x-8)^(1/2),x, algorithm="giac")

[Out]

-1/3*log(abs(-3*x + sqrt(9*x^2 + 6*x - 8) - 1))