Optimal. Leaf size=200 \[ \frac{g e^{-\frac{a}{b}} \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c x}}-\frac{g e^{a/b} \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c x}}+\frac{f \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c x}} \]
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Rubi [A] time = 0.625353, antiderivative size = 242, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5837, 5832, 3317, 3307, 2181} \[ \frac{g e^{-\frac{a}{b}} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c x} \sqrt{c x+1}}-\frac{g e^{a/b} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c x} \sqrt{c x+1}}+\frac{f \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c x} \sqrt{c x+1}} \]
Warning: Unable to verify antiderivative.
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Rule 5837
Rule 5832
Rule 3317
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{1-c x} \sqrt{1+c x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g 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{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^n (c f+g \cosh (x)) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{1-c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (c f (a+b x)^n+g (a+b x)^n \cosh (x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{1-c x} \sqrt{1+c x}}\\ &=\frac{f \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt{1-c x} \sqrt{1+c x}}+\frac{\left (g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{1-c x} \sqrt{1+c x}}\\ &=\frac{f \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt{1-c x} \sqrt{1+c x}}+\frac{\left (g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{1-c x} \sqrt{1+c x}}+\frac{\left (g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{1-c x} \sqrt{1+c x}}\\ &=\frac{f \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt{1-c x} \sqrt{1+c x}}+\frac{e^{-\frac{a}{b}} g \sqrt{1-c^2 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 c^2 \sqrt{1-c x} \sqrt{1+c x}}-\frac{e^{a/b} g \sqrt{1-c^2 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 c^2 \sqrt{1-c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.13196, size = 204, normalized size = 1.02 \[ \frac{e^{-\frac{a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (b g (n+1) \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-b g (n+1) e^{\frac{2 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right )+2 c f e^{a/b} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )}{2 b c^2 (n+1) \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.376, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}{\frac{1}{\sqrt{-cx+1}}}{\frac{1}{\sqrt{cx+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 (g x + f\right )}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt{c x + 1} \sqrt{-c x + 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 x + 1} \sqrt{-c x + 1}{\left (g x + f\right )}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (g x + f\right )}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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