Optimal. Leaf size=46 \[ a x-\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{d}+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.0235829, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5864, 5654, 74} \[ a x-\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{d}+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5864
Rule 5654
Rule 74
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=a x+b \int \cosh ^{-1}(c+d x) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d}-\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d}\\ &=a x-\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{d}+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0492835, size = 61, normalized size = 1.33 \[ a x-\frac{b \left (\sqrt{c+d x-1} \sqrt{c+d x+1}-2 c \sinh ^{-1}\left (\frac{\sqrt{c+d x-1}}{\sqrt{2}}\right )\right )}{d}+b x \cosh ^{-1}(c+d x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 41, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ( dx+c \right ){\rm arccosh} \left (dx+c\right )-\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22302, size = 47, normalized size = 1.02 \begin{align*} a x + \frac{{\left ({\left (d x + c\right )} \operatorname{arcosh}\left (d x + c\right ) - \sqrt{{\left (d x + c\right )}^{2} - 1}\right )} b}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16048, size = 154, normalized size = 3.35 \begin{align*} \frac{a d x +{\left (b d x + b c\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.1967, size = 51, normalized size = 1.11 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{acosh}{\left (c + d x \right )}}{d} + x \operatorname{acosh}{\left (c + d x \right )} - \frac{\sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} & \text{for}\: d \neq 0 \\x \operatorname{acosh}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14232, size = 135, normalized size = 2.93 \begin{align*} -{\left (d{\left (\frac{c \log \left ({\left | -c d -{\left (x{\left | d \right |} - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{\left | d \right |} \right |}\right )}{d{\left | d \right |}} + \frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt{{\left (d x + c\right )}^{2} - 1}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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