Optimal. Leaf size=41 \[ \frac{(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{b} \]
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Rubi [A] time = 0.0158476, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5864, 5654, 74} \[ \frac{(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 5864
Rule 5654
Rule 74
Rubi steps
\begin{align*} \int \cosh ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \cosh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sqrt{-1+a+b x} \sqrt{1+a+b x}}{b}+\frac{(a+b x) \cosh ^{-1}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0362489, size = 56, normalized size = 1.37 \[ x \cosh ^{-1}(a+b x)-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}-2 a \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\rm arccosh} \left (bx+a\right )-\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17374, size = 41, normalized size = 1. \begin{align*} \frac{{\left (b x + a\right )} \operatorname{arcosh}\left (b x + a\right ) - \sqrt{{\left (b x + a\right )}^{2} - 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31516, size = 135, normalized size = 3.29 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.22608, size = 46, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a \operatorname{acosh}{\left (a + b x \right )}}{b} + x \operatorname{acosh}{\left (a + b x \right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} - 1}}{b} & \text{for}\: b \neq 0 \\x \operatorname{acosh}{\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 = 1.17564, size = 126, normalized size = 3.07 \begin{align*} -b{\left (\frac{a \log \left ({\left | -a b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{\left | b \right |} \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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