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

Optimal. Leaf size=22 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]

[Out]

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

[Out]

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

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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)

[Out]

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

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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]

[Out]

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

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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)

[Out]

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

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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")

[Out]

Exception raised: ValueError

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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")

[Out]

-log(-b*x + sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - a)/b

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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)

[Out]

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

[Out]

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