3.1.86 \(\int x^2 \text {arccosh}(a x)^{5/2} \, dx\) [86]

3.1.86.1 Optimal result

Integrand size = 12, antiderivative size = 220 \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\frac {5 x \sqrt {\text {arccosh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arccosh}(a x)}-\frac {5 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{64 a^3}-\frac {5 \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{576 a^3}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{64 a^3}-\frac {5 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{576 a^3} \]

output

1/3*x^3*arccosh(a*x)^(5/2)-5/1728*erf(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)* 
Pi^(1/2)/a^3-5/1728*erfi(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3- 
15/64*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3-15/64*erfi(arccosh(a*x)^(1/2))* 
Pi^(1/2)/a^3-5/9*arccosh(a*x)^(3/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-5/18*x 
^2*arccosh(a*x)^(3/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a+5/6*x*arccosh(a*x)^(1/ 
2)/a^2+5/36*x^3*arccosh(a*x)^(1/2)
 

3.1.86.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.45 \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\frac {\sqrt {3} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {7}{2},-3 \text {arccosh}(a x)\right )+81 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {7}{2},-\text {arccosh}(a x)\right )+\sqrt {-\text {arccosh}(a x)} \left (81 \Gamma \left (\frac {7}{2},\text {arccosh}(a x)\right )+\sqrt {3} \Gamma \left (\frac {7}{2},3 \text {arccosh}(a x)\right )\right )}{648 a^3 \sqrt {-\text {arccosh}(a x)}} \]

input

Integrate[x^2*ArcCosh[a*x]^(5/2),x]
 

output

(Sqrt[3]*Sqrt[ArcCosh[a*x]]*Gamma[7/2, -3*ArcCosh[a*x]] + 81*Sqrt[ArcCosh[ 
a*x]]*Gamma[7/2, -ArcCosh[a*x]] + Sqrt[-ArcCosh[a*x]]*(81*Gamma[7/2, ArcCo 
sh[a*x]] + Sqrt[3]*Gamma[7/2, 3*ArcCosh[a*x]]))/(648*a^3*Sqrt[-ArcCosh[a*x 
]])
 

3.1.86.3 Rubi [A] (verified)

Time = 2.70 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.28, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6299, 6354, 6299, 6330, 6294, 6368, 3042, 3788, 26, 2611, 2633, 2634, 3793, 2009}

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.

\(\displaystyle \int x^2 \text {arccosh}(a x)^{5/2} \, dx\)

\(\Big \downarrow \) 6299

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \int \frac {x^3 \text {arccosh}(a x)^{3/2}}{\sqrt {a x-1} \sqrt {a x+1}}dx\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \int \frac {x \text {arccosh}(a x)^{3/2}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}-\frac {\int x^2 \sqrt {\text {arccosh}(a x)}dx}{2 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 6299

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \int \frac {x \text {arccosh}(a x)^{3/2}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \int \sqrt {\text {arccosh}(a x)}dx}{2 a}\right )}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 6294

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx\right )}{2 a}\right )}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {a^3 x^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {a x}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{2} i \int -\frac {i e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{2} \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)+\frac {1}{2} \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^3}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\int \left (\frac {3 a x}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\cosh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \text {arccosh}(a x)^{5/2}-\frac {5}{6} a \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arccosh}(a x)}-\frac {\frac {3}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {3}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{6 a^3}}{2 a}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}{3 a^2}\right )\)

input

Int[x^2*ArcCosh[a*x]^(5/2),x]
 

output

(x^3*ArcCosh[a*x]^(5/2))/3 - (5*a*((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCo 
sh[a*x]^(3/2))/(3*a^2) + (2*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3 
/2))/a^2 - (3*(x*Sqrt[ArcCosh[a*x]] - ((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/ 
2 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/2)/(2*a)))/(2*a)))/(3*a^2) - ((x^3 
*Sqrt[ArcCosh[a*x]])/3 - ((3*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[P 
i/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8 + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a* 
x]]])/8 + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8)/(6*a^3))/(2*a)) 
)/6
 

3.1.86.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

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]
 

rule 2633

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]
 

rule 2634

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

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]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6299

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]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(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, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6354

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]
 

rule 6368

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]
 

3.1.86.4 Maple [F]

\[\int x^{2} \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}d x\]

input

int(x^2*arccosh(a*x)^(5/2),x)
 

output

int(x^2*arccosh(a*x)^(5/2),x)
 

3.1.86.5 Fricas [F(-2)]

Exception generated. \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^2*arccosh(a*x)^(5/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.1.86.6 Sympy [F(-1)]

Timed out. \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\text {Timed out} \]

input

integrate(x**2*acosh(a*x)**(5/2),x)
 

output

Timed out
 

3.1.86.7 Maxima [F]

\[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\int { x^{2} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}} \,d x } \]

input

integrate(x^2*arccosh(a*x)^(5/2),x, algorithm="maxima")
 

output

integrate(x^2*arccosh(a*x)^(5/2), x)
 

3.1.86.8 Giac [F(-2)]

Exception generated. \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(x^2*arccosh(a*x)^(5/2),x, algorithm="giac")
 

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

3.1.86.9 Mupad [F(-1)]

Timed out. \[ \int x^2 \text {arccosh}(a x)^{5/2} \, dx=\int x^2\,{\mathrm {acosh}\left (a\,x\right )}^{5/2} \,d x \]

input

int(x^2*acosh(a*x)^(5/2),x)
 

output

int(x^2*acosh(a*x)^(5/2), x)