Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\frac {3^{-1-n} d^2 e^{-\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {5 d^2 e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {3^{-1-n} d^2 e^{\frac {3 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+d^2 \text {Int}\left (\frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]
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Not integrable
Time = 0.74 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 (a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}}-\frac {2 c^2 d^2 x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}}+\frac {c^4 d^2 x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}}\right ) \, dx \\ & = d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\left (2 c^2 d^2\right ) \int \frac {x (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx+\left (c^4 d^2\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx \\ & = d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (d^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x^n \cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}}-\frac {\left (2 d^2 \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 \sqrt {d-c^2 d x^2}} \\ & = d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (d^2 \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 )}{b \sqrt {d-c^2 d x^2}}-\frac {\left (d^2 \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 )}{b \sqrt {d-c^2 d x^2}}+\frac {\left (d^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{4} x^n \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )+\frac {3}{4} x^n \cosh \left (\frac {a}{b}-\frac {x}{b}\right )\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = -\frac {d^2 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 )}{\sqrt {d-c^2 d x^2}}+\frac {d^2 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 )}{\sqrt {d-c^2 d x^2}}+d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (d^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{4 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^2 \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 )}{4 b \sqrt {d-c^2 d x^2}} \\ & = -\frac {d^2 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 )}{\sqrt {d-c^2 d x^2}}+\frac {d^2 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 )}{\sqrt {d-c^2 d x^2}}+d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (d^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (d^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^2 \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 )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (3 d^2 \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 )}{8 b \sqrt {d-c^2 d x^2}} \\ & = \frac {3^{-1-n} d^2 e^{-\frac {3 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 {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {5 d^2 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 )}{8 \sqrt {d-c^2 d x^2}}+\frac {5 d^2 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 )}{8 \sqrt {d-c^2 d x^2}}-\frac {3^{-1-n} d^2 e^{\frac {3 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 {3 (a+b \text {arccosh}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+d^2 \int \frac {(a+b \text {arccosh}(c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx \]
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Not integrable
Time = 2.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}}{x}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 3.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]
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