Optimal. Leaf size=211 \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c x}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{2 b c^3 (n+1) \sqrt{1-c x}} \]
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Rubi [A] time = 0.615718, antiderivative size = 250, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {5798, 5781, 3312, 3307, 2181} \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{2 b c^3 (n+1) \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5781
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^2 \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{x^2 \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{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^2(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2} (a+b x)^n+\frac{1}{2} (a+b x)^n \cosh (2 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \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}}{2 b c^3 (1+n) \sqrt{1-c^2 x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 \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}}{2 b c^3 (1+n) \sqrt{1-c^2 x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3 \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}}{2 b c^3 (1+n) \sqrt{1-c^2 x^2}}+\frac{2^{-3-n} e^{-\frac{2 a}{b}} \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{2^{-3-n} e^{\frac{2 a}{b}} \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.787952, size = 212, normalized size = 1. \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (b (n+1) \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b (n+1) e^{\frac{4 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2^{n+2} e^{\frac{2 a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )}{b c^3 (n+1) \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.283, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \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} x^{2}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n} x^{2}}{c^{2} x^{2} - 1}, 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{x^{2} \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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