Integrand size = 12, antiderivative size = 237 \[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {8 x}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^3}{5 \text {arccosh}(a x)^{3/2}}+\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{15 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {24 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{5 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{5 a^3} \]
8/15*x/a^2/arccosh(a*x)^(3/2)-4/5*x^3/arccosh(a*x)^(3/2)+1/15*erf(arccosh( a*x)^(1/2))*Pi^(1/2)/a^3+1/15*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3+3/5*er f(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+3/5*erfi(3^(1/2)*arccos h(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/5*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/a rccosh(a*x)^(5/2)+16/15*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/arccosh(a*x)^(1/2) -24/5*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)
Time = 0.54 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {e^{-3 \text {arccosh}(a x)} \left (-e^{2 \text {arccosh}(a x)} \left (3 e^{\text {arccosh}(a x)} \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x)+e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)-2 \text {arccosh}(a x)^2+2 e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2-2 e^{\text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+2 e^{\text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )-3 \left (\text {arccosh}(a x)+e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)-6 \text {arccosh}(a x)^2+6 e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)^2-6 \sqrt {3} e^{3 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )+6 \sqrt {3} e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+e^{3 \text {arccosh}(a x)} \sinh (3 \text {arccosh}(a x))\right )\right )}{30 a^3 \text {arccosh}(a x)^{5/2}} \]
(-(E^(2*ArcCosh[a*x])*(3*E^ArcCosh[a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a* x) + ArcCosh[a*x] + E^(2*ArcCosh[a*x])*ArcCosh[a*x] - 2*ArcCosh[a*x]^2 + 2 *E^(2*ArcCosh[a*x])*ArcCosh[a*x]^2 - 2*E^ArcCosh[a*x]*(-ArcCosh[a*x])^(5/2 )*Gamma[1/2, -ArcCosh[a*x]] + 2*E^ArcCosh[a*x]*ArcCosh[a*x]^(5/2)*Gamma[1/ 2, ArcCosh[a*x]])) - 3*(ArcCosh[a*x] + E^(6*ArcCosh[a*x])*ArcCosh[a*x] - 6 *ArcCosh[a*x]^2 + 6*E^(6*ArcCosh[a*x])*ArcCosh[a*x]^2 - 6*Sqrt[3]*E^(3*Arc Cosh[a*x])*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -3*ArcCosh[a*x]] + 6*Sqrt[3]*E ^(3*ArcCosh[a*x])*ArcCosh[a*x]^(5/2)*Gamma[1/2, 3*ArcCosh[a*x]] + E^(3*Arc Cosh[a*x])*Sinh[3*ArcCosh[a*x]]))/(30*a^3*E^(3*ArcCosh[a*x])*ArcCosh[a*x]^ (5/2))
Time = 1.93 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6301, 6366, 6295, 6300, 2009, 6368, 3042, 3788, 26, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {6}{5} a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{5/2}}dx-\frac {4 \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{5/2}}dx}{5 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\text {arccosh}(a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {4 \left (\frac {2 \int \frac {1}{\text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\text {arccosh}(a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {6}{5} a \left (\frac {2 \left (-\frac {2 \int \left (-\frac {a x}{4 \sqrt {\text {arccosh}(a x)}}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {a x}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}\right )}{3 a}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}\right )}{3 a}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)+\frac {1}{2} \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {6}{5} a \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\) |
(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) - (4*((-2*x )/(3*a*ArcCosh[a*x]^(3/2)) + (2*((-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt [ArcCosh[a*x]]) + (2*((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/2 + (Sqrt[Pi]*Erf i[Sqrt[ArcCosh[a*x]]])/2))/a))/(3*a)))/(5*a) + (6*a*((-2*x^3)/(3*a*ArcCosh [a*x]^(3/2)) + (2*((-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a *x]]) - (2*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (Sqrt[3*Pi]*Erf[Sqrt [3]*Sqrt[ArcCosh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/8 - (Sqrt [3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8))/a^3))/a))/5
3.2.11.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c /(b*(n + 1)) Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
\[\int \frac {x^{2}}{\operatorname {arccosh}\left (a x \right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]