Integrand size = 8, antiderivative size = 122 \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a} \]
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Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5880, 5951, 5953, 3388, 2211, 2235, 2236} \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5880
Rule 5951
Rule 5953
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {1}{5} (2 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}} \, dx \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}+\frac {4}{15} \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {1}{15} (8 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}} \, dx \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a}+\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a}+\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 e^{-\text {arccosh}(a x)} \left (3 e^{\text {arccosh}(a x)} \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x)+e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)-2 \text {arccosh}(a x)^2+2 e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2-2 e^{\text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+2 e^{\text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )}{15 a \text {arccosh}(a x)^{5/2}} \]
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {-\frac {8 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}{15}+\frac {4 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{15}+\frac {4 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{15}-\frac {4 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x}{15}-\frac {2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}}{5}}{\sqrt {\pi }\, a \operatorname {arccosh}\left (a x \right )^{3}}\) | \(111\) |
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Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
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