Integrand size = 27, antiderivative size = 182 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5952, 3388, 2212} \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {e^{a/b} \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}} \]
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Rule 2212
Rule 3388
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.84 \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (-e^{\frac {2 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{2 c^2 \sqrt {-d (-1+c x) (1+c x)}} \]
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\[\int \frac {x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}}{\sqrt {-c^{2} d \,x^{2}+d}}d x\]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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