Integrand size = 24, antiderivative size = 330 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \]
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Time = 0.45 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5895, 5893, 5884, 5939, 5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a x^2 \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15}{32} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5884
Rule 5887
Rule 5893
Rule 5895
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \text {arccosh}(a x)^{3/2} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.45 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=-\frac {\sqrt {-c (-1+a x) (1+a x)} \left (-105 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+105 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+8 \sqrt {\text {arccosh}(a x)} \left (64 \text {arccosh}(a x)^3+140 \text {arccosh}(a x) \cosh (2 \text {arccosh}(a x))-7 \left (15+16 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )\right )}{3584 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
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\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Timed out} \]
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\[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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