3.758 \(\int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]

[Out]

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

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Rubi [A]  time = 0.0358068, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {93, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]),x]

[Out]

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0480006, size = 88, normalized size = 1.63 \[ \frac{2 \sqrt{a+b x-1} \sqrt{a+b x+1} \tan ^{-1}\left (\frac{\sqrt{a+1} \sqrt{\frac{a+b x-1}{a+b x+1}}}{\sqrt{1-a}}\right )}{\sqrt{1-a^2} \sqrt{-(a+b x-1) (a+b x+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]),x]

[Out]

(2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*ArcTan[(Sqrt[1 + a]*Sqrt[(-1 + a + b*x)/(1 + a + b*x)])/Sqrt[1 - a]])/
(Sqrt[1 - a^2]*Sqrt[-((-1 + a + b*x)*(1 + a + b*x))])

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Maple [C]  time = 0.04, size = 114, normalized size = 2.1 \begin{align*}{\frac{ \left ({\it csgn} \left ( b \right ) \right ) ^{2}}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-bx-a+1}\sqrt{bx+a+1}\sqrt{-{a}^{2}+1}\ln \left ( -2\,{\frac{abx+{a}^{2}-\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}-1}{x}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)*csgn(b)^2*(-a^2+1)^(1/2)*ln(-2*(a*b*x+a^2-(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2)-1)/x)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)/(a-1)/(1+a)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.22638, size = 440, normalized size = 8.15 \begin{align*} \left [-\frac{\sqrt{-a^{2} + 1} \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x + 2 \,{\left (a b x + a^{2} - 1\right )} \sqrt{-a^{2} + 1} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} - 4 \, a^{2} + 2}{x^{2}}\right )}{2 \,{\left (a^{2} - 1\right )}}, \frac{\arctan \left (\frac{{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} \sqrt{b x + a + 1} \sqrt{-b x - a + 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right )}{\sqrt{a^{2} - 1}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1)*s
qrt(b*x + a + 1)*sqrt(-b*x - a + 1) - 4*a^2 + 2)/x^2)/(a^2 - 1), arctan((a*b*x + a^2 - 1)*sqrt(a^2 - 1)*sqrt(b
*x + a + 1)*sqrt(-b*x - a + 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1))/sqrt(a^2 - 1)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- a - b x + 1} \sqrt{a + b x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(-a - b*x + 1)*sqrt(a + b*x + 1)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError