Optimal. Leaf size=11 \[ \frac{\log \left (\sinh ^{-1}(a+b x)\right )}{b} \]
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Rubi [A] time = 0.0751373, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5867, 5673} \[ \frac{\log \left (\sinh ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5867
Rule 5673
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac{\log \left (\sinh ^{-1}(a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0304706, size = 11, normalized size = 1. \[ \frac{\log \left (\sinh ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it Arcsinh} \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname{arsinh}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.63541, size = 77, normalized size = 7. \begin{align*} \frac{\log \left (\log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.17822, size = 22, normalized size = 2. \begin{align*} \begin{cases} \frac{\log{\left (\operatorname{asinh}{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{a^{2} + 1} \operatorname{asinh}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28502, size = 42, normalized size = 3.82 \begin{align*} \frac{\log \left ({\left | \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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