Optimal. Leaf size=30 \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]
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Rubi [A] time = 0.0141884, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5653, 261} \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5653
Rule 261
Rubi steps
\begin{align*} \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=a x+b \int \sinh ^{-1}(c x) \, dx\\ &=a x+b x \sinh ^{-1}(c x)-(b c) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx\\ &=a x-\frac{b \sqrt{1+c^2 x^2}}{c}+b x \sinh ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.0081827, size = 30, normalized size = 1. \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 31, normalized size = 1. \begin{align*} ax+{\frac{b}{c} \left ({\it Arcsinh} \left ( cx \right ) cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22996, size = 41, normalized size = 1.37 \begin{align*} a x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34865, size = 95, normalized size = 3.17 \begin{align*} \frac{b c x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a c x - \sqrt{c^{2} x^{2} + 1} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.151141, size = 26, normalized size = 0.87 \begin{align*} a x + b \left (\begin{cases} x \operatorname{asinh}{\left (c x \right )} - \frac{\sqrt{c^{2} x^{2} + 1}}{c} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46019, size = 55, normalized size = 1.83 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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