\(\int x^3 \text {arccosh}(a x)^{3/2} \, dx\) [79]

Optimal result

Integrand size = 12, antiderivative size = 209 \[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4} \]

[Out]

-3/32*arccosh(a*x)^(3/2)/a^4+1/4*x^4*arccosh(a*x)^(3/2)-3/256*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)
/a^4+3/256*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-3/2048*erf(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4
+3/2048*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4-9/64*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(1/2)/a^3-3/32
*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5884, 5939, 5893, 5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}-\frac {9 x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{64 a^3}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{32 a} \]

[In]

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

[Out]

(-9*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(64*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[Arc
Cosh[a*x]])/(32*a) - (3*ArcCosh[a*x]^(3/2))/(32*a^4) + (x^4*ArcCosh[a*x]^(3/2))/4 - (3*Sqrt[Pi]*Erf[2*Sqrt[Arc
Cosh[a*x]]])/(2048*a^4) - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(128*a^4) + (3*Sqrt[Pi]*Erfi[2*Sqrt[A
rcCosh[a*x]]])/(2048*a^4) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(128*a^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

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 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[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 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(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*Arc
Cosh[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]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(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] + (Dist[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] - Dist[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] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}+\frac {3}{64} \int \frac {x^3}{\sqrt {\text {arccosh}(a x)}} \, dx-\frac {9 \int \frac {x^2 \sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a^4}-\frac {9 \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 a^3}+\frac {9 \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx}{128 a^2} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{64 a^4}+\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{128 a^4} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{128 a^4} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{1024 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{1024 a^4}-\frac {3 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{512 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{256 a^4} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{512 a^4}-\frac {3 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{256 a^4}+\frac {3 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {9 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{512 a^4}+\frac {9 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{512 a^4} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{512 a^4}-\frac {9 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{256 a^4} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.48 \[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=\frac {\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-4 \text {arccosh}(a x)\right )+8 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-2 \text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (8 \sqrt {2} \Gamma \left (\frac {5}{2},2 \text {arccosh}(a x)\right )+\Gamma \left (\frac {5}{2},4 \text {arccosh}(a x)\right )\right )}{512 a^4 \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -4*ArcCosh[a*x]] + 8*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -2*ArcCosh[a*x]] +
Sqrt[ArcCosh[a*x]]*(8*Sqrt[2]*Gamma[5/2, 2*ArcCosh[a*x]] + Gamma[5/2, 4*ArcCosh[a*x]]))/(512*a^4*Sqrt[ArcCosh
[a*x]])

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-32 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+24 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-3 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{256 \sqrt {\pi }\, a^{4}}-\frac {-512 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}+192 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+512 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-96 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -64 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-3 \pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{2048 \sqrt {\pi }\, a^{4}}\)	\(242\)

[In]

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

[Out]

-1/256*2^(1/2)*(-32*2^(1/2)*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2+24*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)
^(1/2)*(a*x-1)^(1/2)*a*x+16*2^(1/2)*arccosh(a*x)^(3/2)*Pi^(1/2)+3*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-3*Pi*erfi
(2^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4-1/2048*(-512*arccosh(a*x)^(3/2)*Pi^(1/2)*a^4*x^4+192*arccosh(a*x)^(
1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*x^3+512*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2-96*arccosh(a*x)^(1/2
)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-64*arccosh(a*x)^(3/2)*Pi^(1/2)+3*Pi*erf(2*arccosh(a*x)^(1/2))-3*Pi*
erfi(2*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

Integral(x**3*acosh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^3*arccosh(a*x)^(3/2), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3*acosh(a*x)^(3/2), x)