Integrand size = 12, antiderivative size = 300 \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5} \]
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Time = 0.59 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5886, 5951, 5885, 3388, 2211, 2235, 2236} \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {32 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a^3 \sqrt {\text {arccosh}(a x)}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}-\frac {40 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \sqrt {\text {arccosh}(a x)}}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5885
Rule 5886
Rule 5951
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {8 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}} \, dx \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {20}{3} \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx-\frac {16 \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx}{5 a^2} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}+\frac {32 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}-\frac {3 \cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}-\frac {40 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{8 \sqrt {x}}-\frac {9 \cosh (3 x)}{16 \sqrt {x}}-\frac {5 \cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{3 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}-\frac {8 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{6 a^5}-\frac {24 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}-\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{6 a^5}+\frac {5 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^5}-\frac {12 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}-\frac {12 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{4 a^5}+\frac {15 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{4 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}-\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^5}-\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a^5}+\frac {5 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^5}-\frac {24 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^5}-\frac {24 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a^5}+\frac {15 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5} \\ \end{align*}
Time = 1.28 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.25 \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {-6 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-2 e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-2 e^{\text {arccosh}(a x)} \text {arccosh}(a x)+4 e^{-\text {arccosh}(a x)} \text {arccosh}(a x)^2-4 e^{\text {arccosh}(a x)} \text {arccosh}(a x)^2+4 (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-4 \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )-5 \text {arccosh}(a x) \left (e^{-5 \text {arccosh}(a x)} (1-10 \text {arccosh}(a x))+e^{5 \text {arccosh}(a x)} (1+10 \text {arccosh}(a x))+10 \sqrt {5} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )+10 \sqrt {5} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )\right )-9 e^{-3 \text {arccosh}(a x)} \left (\text {arccosh}(a x)+e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)-6 \text {arccosh}(a x)^2+6 e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)^2-6 \sqrt {3} e^{3 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )+6 \sqrt {3} e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+e^{3 \text {arccosh}(a x)} \sinh (3 \text {arccosh}(a x))\right )-3 \sinh (5 \text {arccosh}(a x))}{120 a^5 \text {arccosh}(a x)^{5/2}} \]
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\[\int \frac {x^{4}}{\operatorname {arccosh}\left (a x \right )^{\frac {7}{2}}}d x\]
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Exception generated. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
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