Optimal. Leaf size=43 \[ \frac{\sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c x}} \]
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Rubi [A] time = 0.207832, antiderivative size = 56, normalized size of antiderivative = 1.3, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {5713, 5676} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c^2 x^2}} \]
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 )^n}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0389282, size = 56, normalized size = 1.3 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 53, normalized size = 1.2 \begin{align*}{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{1+n}}{cb \left ( 1+n \right ) }\sqrt{cx-1}\sqrt{cx+1}{\frac{1}{\sqrt{- \left ( cx-1 \right ) \left ( cx+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt{-c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.60406, size = 487, normalized size = 11.33 \begin{align*} \frac{{\left (\sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + \sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} a\right )} \cosh \left (n \log \left (b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a\right )\right ) +{\left (\sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + \sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} a\right )} \sinh \left (n \log \left (b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a\right )\right )}{b c n -{\left (b c^{3} n + b c^{3}\right )} x^{2} + b c} \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 )^{n}}{\sqrt{- \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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