Integrand size = 12, antiderivative size = 160 \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c} \]
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Time = 0.52 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5879, 5915, 5953, 3388, 2211, 2236, 2235} \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c}+\frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5879
Rule 5915
Rule 5953
Rubi steps \begin{align*} \text {integral}& = x (a+b \text {arccosh}(c x))^{5/2}-\frac {1}{2} (5 b c) \int \frac {x (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}+\frac {1}{4} \left (15 b^2\right ) \int \sqrt {a+b \text {arccosh}(c x)} \, dx \\ & = \frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {1}{8} \left (15 b^3 c\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx \\ & = \frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 c} \\ & = \frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 c}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 c} \\ & = \frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 c}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 c} \\ & = \frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(452\) vs. \(2(160)=320\).
Time = 1.67 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.82 \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {4 b \sqrt {a+b \text {arccosh}(c x)} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a-5 b \text {arccosh}(c x))+b c x \left (15+4 \text {arccosh}(c x)^2\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )-\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 a b \left (-12 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{16 c} \]
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\[\int \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \]
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