\(\int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx\) [104]

Optimal result

Integrand size = 12, antiderivative size = 172 \[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {4 x^2}{a^2 \sqrt {\text {arccosh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a^4}-\frac {\sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a^4}+\frac {2 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a^4}+\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a^4} \] Output:

-2/3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+4*x^2/a^2/arccos 
h(a*x)^(1/2)-16/3*x^4/arccosh(a*x)^(1/2)-2/3*Pi^(1/2)*erf(2*arccosh(a*x)^( 
1/2))/a^4-1/3*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a^4+2/3*Pi^ 
(1/2)*erfi(2*arccosh(a*x)^(1/2))/a^4+1/3*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arc 
cosh(a*x)^(1/2))/a^4
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-4 \text {arccosh}(a x) \left (e^{-4 \text {arccosh}(a x)}+e^{4 \text {arccosh}(a x)}-2 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-2 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )-2 \left (2 \text {arccosh}(a x) \left (e^{-2 \text {arccosh}(a x)}+e^{2 \text {arccosh}(a x)}-\sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )-\sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )+\sinh (2 \text {arccosh}(a x))\right )-\sinh (4 \text {arccosh}(a x))}{12 a^4 \text {arccosh}(a x)^{3/2}} \] Input:

Integrate[x^3/ArcCosh[a*x]^(5/2),x]
 

Output:

(-4*ArcCosh[a*x]*(E^(-4*ArcCosh[a*x]) + E^(4*ArcCosh[a*x]) - 2*Sqrt[-ArcCo 
sh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - 2*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*A 
rcCosh[a*x]]) - 2*(2*ArcCosh[a*x]*(E^(-2*ArcCosh[a*x]) + E^(2*ArcCosh[a*x] 
) - Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] - Sqrt[2]*Sqrt 
[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a*x]]) + Sinh[2*ArcCosh[a*x]]) - Sinh[ 
4*ArcCosh[a*x]])/(12*a^4*ArcCosh[a*x]^(3/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.46, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6301, 6366, 6302, 5971, 27, 2009, 3042, 26, 3789, 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.

\(\displaystyle \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6301

\(\displaystyle \frac {8}{3} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx-\frac {2 \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx}{a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6366

\(\displaystyle -\frac {2 \left (\frac {4 \int \frac {x}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \int \frac {x^3}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6302

\(\displaystyle -\frac {2 \left (\frac {4 \int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \int \frac {a^3 x^3 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {8}{3} a \left (\frac {8 \int \left (\frac {\sinh (2 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (4 \text {arccosh}(a x))}{8 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 \left (\frac {4 \int \frac {\sinh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {8}{3} a \left (\frac {8 \int \left (\frac {\sinh (2 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (4 \text {arccosh}(a x))}{8 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 \left (\frac {2 \int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {2 \int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int -\frac {i \sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \int \frac {\sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3789

\(\displaystyle -\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2611

\(\displaystyle -\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (i \int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}\right )}{a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

Input:

Int[x^3/ArcCosh[a*x]^(5/2),x]
 

Output:

(-2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (2*((-2*x 
^2)/(a*Sqrt[ArcCosh[a*x]]) - ((2*I)*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ 
ArcCosh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/a^3)) 
/a + (8*a*((-2*x^4)/(a*Sqrt[ArcCosh[a*x]]) + (8*(-1/32*(Sqrt[Pi]*Erf[2*Sqr 
t[ArcCosh[a*x]]]) - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt 
[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCos 
h[a*x]]])/8))/a^5))/3
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3789

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]
 

rule 5971

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]
 

rule 6301

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]
 

rule 6302

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]
 

rule 6366

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(132)=264\).

Time = 0.41 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.61

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-2 \operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{6 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{2}}-\frac {16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}+2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-\sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-2 \operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }}{3 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{2}}\)	\(277\)

Input:

int(x^3/arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6*2^(1/2)*(4*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+2^(1/2)*arccos 
h(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+2*arccosh(a*x)^2*Pi* 
erf(2^(1/2)*arccosh(a*x)^(1/2))-2*arccosh(a*x)^2*Pi*erfi(2^(1/2)*arccosh(a 
*x)^(1/2))-2*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2))/Pi^(1/2)/a^4/arccosh(a*x 
)^2-1/3*(16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^4*x^4+2*arccosh(a*x)^(1/2)*Pi^(1 
/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*x^3-16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2 
*x^2-arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+2*arccosh 
(a*x)^2*Pi*erf(2*arccosh(a*x)^(1/2))-2*arccosh(a*x)^2*Pi*erfi(2*arccosh(a* 
x)^(1/2))+2*arccosh(a*x)^(3/2)*Pi^(1/2))/Pi^(1/2)/a^4/arccosh(a*x)^2
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/arccosh(a*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^{3}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**3/acosh(a*x)**(5/2),x)
 

Output:

Integral(x**3/acosh(a*x)**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3/arccosh(a*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^3/arccosh(a*x)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/arccosh(a*x)^(5/2),x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \] Input:

int(x^3/acosh(a*x)^(5/2),x)
 

Output:

int(x^3/acosh(a*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^3}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {\frac {8 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{4}}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}}{3}-2 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right )-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{3}}{\mathit {acosh} \left (a x \right )^{2} a} \] Input:

int(x^3/acosh(a*x)^(5/2),x)
 

Output:

(2*(4*acosh(a*x)**2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**4 
)/(acosh(a*x)**2*a**2*x**2 - acosh(a*x)**2),x)*a**2 - 3*acosh(a*x)**2*int( 
(sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**2)/(acosh(a*x)**2*a**2*x* 
*2 - acosh(a*x)**2),x) - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**3 
))/(3*acosh(a*x)**2*a)