Integrand size = 12, antiderivative size = 244 \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4} \]
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Time = 0.50 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5886, 5951, 5885, 3388, 2211, 2235, 2236} \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {16 x \sqrt {a x-1} \sqrt {a x+1}}{5 a^3 \sqrt {\text {arccosh}(a x)}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a x-1} \sqrt {a x+1}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}} \]
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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^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {6 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}} \, dx \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {64}{15} \int \frac {x^3}{\text {arccosh}(a x)^{3/2}} \, dx-\frac {8 \int \frac {x}{\text {arccosh}(a x)^{3/2}} \, dx}{5 a^2} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^4}-\frac {128 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}-\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {8 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^4}-\frac {8 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{15 a^4}-\frac {16 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^4}-\frac {16 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{5 a^4} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{5 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{15 a^4} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {e^{-4 \text {arccosh}(a x)} \left (3-3 e^{8 \text {arccosh}(a x)}-8 \text {arccosh}(a x)-8 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)+64 \text {arccosh}(a x)^2-64 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)^2+128 e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-8 e^{2 \text {arccosh}(a x)} \left (3 a e^{2 \text {arccosh}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x)+e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)-4 \text {arccosh}(a x)^2+4 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^2-4 \sqrt {2} e^{2 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+4 \sqrt {2} e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )-128 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )}{120 a^4 \text {arccosh}(a x)^{5/2}} \]
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Time = 1.49 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {\sqrt {2}\, \left (-16 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -4 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-3 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+2 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\right )}{30 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{3}}+\frac {-128 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} a^{3} x^{3}-16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}-6 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+64 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} a x +16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+16 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }}{15 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{3}}\) | \(366\) |
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Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]
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