Integrand size = 24, antiderivative size = 330 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \]
1/2*x*arccosh(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+5/16*arccosh(a*x)^(3/2)*(-a^ 2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-5/8*a*x^2*arccosh(a*x)^(3/2 )*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/7*arccosh(a*x)^(7/2)* (-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/512*erf(2^(1/2)*arcc osh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x +1)^(1/2)-15/512*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c *x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/32*x*(-a^2*c*x^2+c)^(1/2)*a rccosh(a*x)^(1/2)
Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.45 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=-\frac {\sqrt {-c (-1+a x) (1+a x)} \left (-105 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+105 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+8 \sqrt {\text {arccosh}(a x)} \left (64 \text {arccosh}(a x)^3+140 \text {arccosh}(a x) \cosh (2 \text {arccosh}(a x))-7 \left (15+16 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )\right )}{3584 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
-1/3584*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(-105*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt [ArcCosh[a*x]]] + 105*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 8*Sqrt [ArcCosh[a*x]]*(64*ArcCosh[a*x]^3 + 140*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] - 7*(15 + 16*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
Result contains complex when optimal does not.
Time = 2.60 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.78, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6310, 6299, 6308, 6354, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}
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.
| \(\displaystyle \int \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2} \, dx\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \int x \text {arccosh}(a x)^{3/2}dx}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 6299 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {\text {arccosh}(a x)}}dx}{4 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int -\frac {i \sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \int \frac {\sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (i \int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a^3}+\frac {\text {arccosh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}\) |
(x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCos h[a*x]^(7/2))/(7*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (5*a*Sqrt[c - a^2*c*x^2 ]*((x^2*ArcCosh[a*x]^(3/2))/2 - (3*a*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt [ArcCosh[a*x]])/(2*a^2) + ArcCosh[a*x]^(3/2)/(3*a^3) + ((I/8)*((-1/2*I)*Sq rt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*S qrt[ArcCosh[a*x]]]))/a^3))/4))/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
3.4.89.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[x*(a + b*ArcCosh[c*x])^ (n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]