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 = 0.0206604, 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, Rules used = {688, 208} \[ -\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 688
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx &=\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 a^2 b^3-4 \left (1+a^2\right ) b^3+4 b^3 x^2} \, dx,x,\sqrt{1+a^2+2 a b x+b^2 x^2}\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{1+a^2+2 a b x+b^2 x^2}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0081073, size = 19, normalized size = 0.7 \[ -\frac{\tanh ^{-1}\left (\sqrt{(a+b x)^2+1}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 24, normalized size = 0.9 \begin{align*} -{\frac{1}{b}{\it Artanh} \left ({\frac{1}{\sqrt{ \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.5732, size = 19, normalized size = 0.7 \begin{align*} -\frac{\operatorname{arsinh}\left (\frac{1}{{\left | b x + a \right |}}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05115, size = 157, normalized size = 5.81 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23572, size = 120, normalized size = 4.44 \begin{align*} \frac{\log \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 |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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