Optimal. Leaf size=27 \[ -\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b} \]
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Rubi [A] time = 0.0660912, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]),x]
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Rubi in Sympy [A] time = 18.9949, size = 26, normalized size = 0.96 \[ - \frac{\operatorname{atanh}{\left (\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(b**2*x**2+2*a*b*x+a**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.020756, size = 32, normalized size = 1.19 \[ \frac{\log (a+b x)}{b}-\frac{\log \left (\sqrt{(a+b x)^2+1}+1\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Maple [A] time = 0.011, size = 24, normalized size = 0.9 \[ -{\frac{1}{b}{\it Artanh} \left ({\frac{1}{\sqrt{ \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+1}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 0.757424, size = 19, normalized size = 0.7 \[ -\frac{\operatorname{arsinh}\left (\frac{1}{{\left | b x + a \right |}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216567, size = 89, normalized size = 3.3 \[ -\frac{\log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) - \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(b**2*x**2+2*a*b*x+a**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.241197, size = 120, normalized size = 4.44 \[ \frac{{\rm ln}\left (\frac{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} - 2 \,{\left | b \right |} \right |}}{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} + 2 \,{\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)),x, algorithm="giac")
[Out]