Integrand size = 12, antiderivative size = 228 \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5} \]
[Out]
Time = 0.60 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5886, 5951, 5887, 5556, 3389, 2211, 2235, 2236} \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5886
Rule 5887
Rule 5951
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {8 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}+\frac {100}{3} \int \frac {x^4}{\sqrt {\text {arccosh}(a x)}} \, dx-\frac {16 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx}{a^2} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {16 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {x}}+\frac {3 \sinh (3 x)}{16 \sqrt {x}}+\frac {\sinh (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}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}+\frac {25 \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^5}-\frac {4 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}-\frac {4 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sinh (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}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {25 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{24 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{24 a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{8 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{8 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {25 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {4 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^5}+\frac {4 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^5}-\frac {25 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^5}-\frac {25 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{4 a^5}+\frac {25 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{4 a^5} \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.25 \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-5 e^{-5 \text {arccosh}(a x)} \text {arccosh}(a x)-5 e^{5 \text {arccosh}(a x)} \text {arccosh}(a x)-5 \sqrt {5} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )-2 \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+e^{-\text {arccosh}(a x)} \text {arccosh}(a x)+e^{\text {arccosh}(a x)} \text {arccosh}(a x)+(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )+5 \sqrt {5} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )-3 \left (3 e^{-3 \text {arccosh}(a x)} \text {arccosh}(a x)+3 e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)+3 \sqrt {3} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-3 \sqrt {3} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+\sinh (3 \text {arccosh}(a x))\right )-\sinh (5 \text {arccosh}(a x))}{24 a^5 \text {arccosh}(a x)^{3/2}} \]
[In]
[Out]
\[\int \frac {x^{4}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]
[In]
[Out]