Integrand size = 27, antiderivative size = 578 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d e^{-\frac {a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d e^{a/b} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.37 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5952, 5556, 3388, 2212} \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {d 5^{-n-1} e^{-\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 3^{-n} e^{-\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d e^{-\frac {a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d e^{a/b} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 3^{-n} e^{\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 5^{-n-1} e^{\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 2212
Rule 3388
Rule 5556
Rule 5952
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{16} x^n \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )-\frac {3}{16} x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {1}{8} x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\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 )}{8 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\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 )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\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 )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b 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} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{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} (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 )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d e^{a/b} \sqrt {d-c^2 d x^2} (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 )}{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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 1.65 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.87 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {15^{-1-n} d^2 e^{-\frac {5 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-3 n} \left (2\ 15^{1+n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-3^{1+n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+3\ 5^{1+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-2\ 15^{1+n} e^{\frac {4 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )+5^{1+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-4\ 5^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{32 c^2 \sqrt {d-c^2 d x^2}} \]
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\[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]
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\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \]
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Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \]
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\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \]
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Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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