Optimal. Leaf size=379 \[ \frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.80871, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {5798, 5781, 3312, 3307, 2181} \[ \frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5781
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^3(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{4} (a+b x)^n \cosh (x)+\frac{1}{4} (a+b x)^n \cosh (3 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \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 )}{4 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \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 )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \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 )}{8 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{3^{-1-n} e^{-\frac{3 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{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 e^{-\frac{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{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 e^{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{a+b \cosh ^{-1}(c x)}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3^{-1-n} e^{\frac{3 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{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 c^4 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.07402, size = 291, normalized size = 0.77 \[ -\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (3^{n+2} e^{\frac{4 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right )-\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \left (\left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3^{n+2} e^{\frac{2 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-e^{\frac{6 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )\right )}{8 c^4 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.335, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}\, 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 (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n} x^{3}}{\sqrt{-c^{2} d x^{2} + d}}\,{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} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n} x^{3}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[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]