Optimal. Leaf size=211 \[ d \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{d-c^2 d x^2}},x\right )-\frac{d e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt{d-c^2 d x^2}}+\frac{d e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 1.0825, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{d-c^2 d x^2} \int \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 x \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \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{a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{e^{a/b} \sqrt{d-c^2 d x^2} \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{a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.226385, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}}{x}\sqrt{-{c}^{2}d{x}^{2}+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{n}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] 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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