Optimal. Leaf size=239 \[ \frac{g e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}-\frac{g e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}+\frac{f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.470914, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5836, 5832, 3317, 3307, 2181} \[ \frac{g e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}-\frac{g e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}+\frac{f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5836
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^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \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^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \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^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \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^2 x^2}}\\ &=\frac{f \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}}+\frac{\left (g \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{f \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}}+\frac{\left (g \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (g \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{f \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}}+\frac{e^{-\frac{a}{b}} g \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{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}-\frac{e^{a/b} g \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{a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.632461, size = 204, normalized size = 0.85 \[ \frac{e^{-\frac{a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (b g (n+1) \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-b g (n+1) e^{\frac{2 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right )+2 c f e^{a/b} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )}{2 b c^2 (n+1) \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.322, 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{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{n} \left (f + g x\right )}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]