Optimal. Leaf size=51 \[ -\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2+2 b^2 x \]
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Rubi [A] time = 0.156815, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5654, 5718, 8} \[ -\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2+2 b^2 x \]
Antiderivative was successfully verified.
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Rule 5654
Rule 5718
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \cosh ^{-1}(c x)\right )^2-(2 b c) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2+\left (2 b^2\right ) \int 1 \, dx\\ &=2 b^2 x-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.0870446, size = 84, normalized size = 1.65 \[ x \left (a^2+2 b^2\right )-\frac{2 a b \sqrt{c x-1} \sqrt{c x+1}}{c}+\frac{2 b \cosh ^{-1}(c x) \left (a c x-b \sqrt{c x-1} \sqrt{c x+1}\right )}{c}+b^2 x \cosh ^{-1}(c x)^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 78, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ( cx{a}^{2}+{b}^{2} \left ( \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cx-2\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+2\,cx \right ) +2\,ab \left ( cx{\rm arccosh} \left (cx\right )-\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17358, size = 97, normalized size = 1.9 \begin{align*} b^{2} x \operatorname{arcosh}\left (c x\right )^{2} + 2 \, b^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} - 1} \operatorname{arcosh}\left (c x\right )}{c}\right )} + a^{2} x + \frac{2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} a b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71647, size = 212, normalized size = 4.16 \begin{align*} \frac{b^{2} c x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} +{\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt{c^{2} x^{2} - 1} a b + 2 \,{\left (a b c x - \sqrt{c^{2} x^{2} - 1} b^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.363591, size = 88, normalized size = 1.73 \begin{align*} \begin{cases} a^{2} x + 2 a b x \operatorname{acosh}{\left (c x \right )} - \frac{2 a b \sqrt{c^{2} x^{2} - 1}}{c} + b^{2} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac{2 b^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} & \text{for}\: c \neq 0 \\x \left (a + \frac{i \pi b}{2}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32592, size = 150, normalized size = 2.94 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} a b +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} - 1} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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