Optimal. Leaf size=22 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]
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Rubi [A] time = 0.0318357, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]),x]
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Rubi in Sympy [A] time = 6.30871, size = 24, normalized size = 1.09 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x + 1}}{\sqrt{a + b x - 1}} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a-1)**(1/2)/(b*x+a+1)**(1/2),x)
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Mathematica [A] time = 0.0143631, size = 22, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]),x]
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Maple [B] time = 0.008, size = 94, normalized size = 4.3 \[{1\sqrt{ \left ( bx+a-1 \right ) \left ( bx+a+1 \right ) }\ln \left ({1 \left ({\frac{b \left ( a-1 \right ) }{2}}+{\frac{b \left ( 1+a \right ) }{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( b \left ( a-1 \right ) +b \left ( 1+a \right ) \right ) x+ \left ( a-1 \right ) \left ( 1+a \right ) } \right ){\frac{1}{\sqrt{bx+a-1}}}{\frac{1}{\sqrt{bx+a+1}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a-1)^(1/2)/(b*x+a+1)^(1/2),x)
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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, algorithm="maxima")
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Fricas [A] time = 0.227463, size = 42, normalized size = 1.91 \[ -\frac{\log \left (-b x + \sqrt{b x + a + 1} \sqrt{b x + a - 1} - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x - 1} \sqrt{a + b x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a-1)**(1/2)/(b*x+a+1)**(1/2),x)
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GIAC/XCAS [A] time = 0.268985, size = 35, normalized size = 1.59 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + a + 1} + \sqrt{b x + a - 1} \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)),x, algorithm="giac")
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