Optimal. Leaf size=53 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.195441, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5713, 5676} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5676
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0497908, size = 53, normalized size = 1. \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 149, normalized size = 2.8 \begin{align*}{{a}^{2}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{3}}{3\,cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{- \left ( cx-1 \right ) \left ( cx+1 \right ) d}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{ab \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{- \left ( cx-1 \right ) \left ( cx+1 \right ) d}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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