Optimal. Leaf size=35 \[ a x-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c}+b x \cosh ^{-1}(c x) \]
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Rubi [A] time = 0.0143607, antiderivative size = 35, 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 = {5654, 74} \[ a x-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c}+b x \cosh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5654
Rule 74
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=a x+b \int \cosh ^{-1}(c x) \, dx\\ &=a x+b x \cosh ^{-1}(c x)-(b c) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=a x-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c}+b x \cosh ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.0207335, size = 35, normalized size = 1. \[ a x-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c}+b x \cosh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 1. \begin{align*} ax+{\frac{b}{c} \left ( cx{\rm arccosh} \left (cx\right )-\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10648, size = 41, normalized size = 1.17 \begin{align*} a x + \frac{{\left (c x \operatorname{arcosh}\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 = 2.49282, size = 95, normalized size = 2.71 \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 = 0.166587, size = 31, normalized size = 0.89 \begin{align*} a x + b \left (\begin{cases} x \operatorname{acosh}{\left (c x \right )} - \frac{\sqrt{c^{2} x^{2} - 1}}{c} & \text{for}\: c \neq 0 \\\frac{i \pi x}{2} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29573, size = 55, normalized size = 1.57 \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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