Optimal. Leaf size=450 \[ -\frac{d 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.563549, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5713, 5701, 3312, 3307, 2181} \[ -\frac{d 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5701
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \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 (-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 \sinh ^4(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c \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{3}{8} (a+b x)^n-\frac{1}{2} (a+b x)^n \cosh (2 x)+\frac{1}{8} (a+b x)^n \cosh (4 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \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 (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c \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 (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4^{-3-n} d e^{-\frac{4 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{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2^{-3-n} d e^{-\frac{2 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{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2^{-3-n} d e^{\frac{2 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{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4^{-3-n} d e^{\frac{4 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{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 2.0866, size = 384, normalized size = 0.85 \[ \frac{d^2 4^{-n-3} e^{-\frac{4 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (b \left (-2^{n+3}\right ) (n+1) e^{\frac{2 a}{b}} \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 2^{n+3} (n+1) e^{\frac{6 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b (n+1) e^{\frac{8 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b (n+1) \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^n \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )^{2 n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3\ 2^{2 n+3} e^{\frac{4 a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n}\right )}{b c (n+1) \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.216, size = 0, normalized size = 0. \begin{align*} \int \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}\,{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^{2} + d\right )}^{\frac{3}{2}}{\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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