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

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])])/Sq
rt[1 - a^2]

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Rubi [A]  time = 0.111688, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{\sqrt{1-a^2}} \]

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])])/Sq
rt[1 - a^2]

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Rubi in Sympy [A]  time = 9.85343, size = 51, normalized size = 0.94 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{- a + 1} \sqrt{a + b x + 1}}{\sqrt{a + 1} \sqrt{- a - b x + 1}} \right )}}{\sqrt{- a + 1} \sqrt{a + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-b*x-a+1)**(1/2)/(b*x+a+1)**(1/2),x)

[Out]

-2*atanh(sqrt(-a + 1)*sqrt(a + b*x + 1)/(sqrt(a + 1)*sqrt(-a - b*x + 1)))/(sqrt(
-a + 1)*sqrt(a + 1))

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Mathematica [C]  time = 0.171886, size = 107, normalized size = 1.98 \[ -\frac{i \sqrt{a+b x-1} \sqrt{a+b x+1} \log \left (\frac{2 \sqrt{a+b x-1} \sqrt{a+b x+1}}{x}+\frac{2 i \left (a^2+a b x-1\right )}{\sqrt{1-a^2} x}\right )}{\sqrt{1-a^2} \sqrt{-(a+b x-1) (a+b x+1)}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.059, size = 114, normalized size = 2.1 \[{\frac{ \left ({\it csgn} \left ( b \right ) \right ) ^{2}}{ \left ( a-1 \right ) \left ( 1+a \right ) }\sqrt{-bx-a+1}\sqrt{bx+a+1}\ln \left ( -2\,{\frac{abx+{a}^{2}-\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}-1}{x}} \right ) \sqrt{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236267, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \,{\left (a^{4} +{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} -{\left ({\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 4 \, a^{2} + 2\right )} \sqrt{-a^{2} + 1}}{x^{2}}\right )}{2 \, \sqrt{-a^{2} + 1}}, -\frac{\arctan \left (\frac{a b x + a^{2} - 1}{\sqrt{a^{2} - 1} \sqrt{b x + a + 1} \sqrt{-b x - a + 1}}\right )}{\sqrt{a^{2} - 1}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x),x, algorithm="fricas")

[Out]

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

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

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x),x, algorithm="giac")

[Out]

Exception raised: TypeError