Optimal. Leaf size=578 \[ -\frac{d 5^{-n-1} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 3^{-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d e^{a/b} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 3^{-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 5^{-n-1} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.825303, antiderivative size = 578, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5798, 5781, 5448, 3307, 2181} \[ -\frac{d 5^{-n-1} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 3^{-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d e^{a/b} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 3^{-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 5^{-n-1} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5781
Rule 5448
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \sinh ^4(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{8} (a+b x)^n \cosh (x)-\frac{3}{16} (a+b x)^n \cosh (3 x)+\frac{1}{16} (a+b x)^n \cosh (5 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (5 x) \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5^{-1-n} d e^{-\frac{5 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{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3^{-n} d e^{-\frac{3 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{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d 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 )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d 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 )}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3^{-n} d e^{\frac{3 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{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5^{-1-n} d e^{\frac{5 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{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 2.09443, size = 500, normalized size = 0.87 \[ -\frac{d^2 15^{-n-1} e^{-\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-3 n} \left (\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \left (-3^{n+1} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3\ 5^{n+1} e^{\frac{2 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-2\ 15^{n+1} e^{\frac{4 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-4\ 5^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+5^{n+1} e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3^{n+1} e^{\frac{10 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )+2\ 15^{n+1} e^{\frac{6 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )}{32 c^2 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int x \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}\, 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{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (c^{2} d x^{3} - d x\right )} \sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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