\(\int x^2 \text {arccosh}(a x)^{3/2} \, dx\) [80]

Optimal result

Integrand size = 12, antiderivative size = 189 \[ \int x^2 \text {arccosh}(a x)^{3/2} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3} \]

[Out]

1/3*x^3*arccosh(a*x)^(3/2)-1/288*erf(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/288*erfi(3^(1/2)*arcco
sh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-3/32*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3+3/32*erfi(arccosh(a*x)^(1/2))*Pi
^(1/2)/a^3-1/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(1/2)/a^3-1/6*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(
a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5884, 5939, 5915, 5881, 3389, 2211, 2235, 2236, 5887, 5556} \[ \int x^2 \text {arccosh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3}-\frac {\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{3 a^3}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{6 a} \]

[In]

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

[Out]

-1/3*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/a^3 - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*
x]])/(6*a) + (x^3*ArcCosh[a*x]^(3/2))/3 - (3*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(32*a^3) - (Sqrt[Pi/3]*Erf[Sqrt
[3]*Sqrt[ArcCosh[a*x]]])/(96*a^3) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*
Sqrt[ArcCosh[a*x]]])/(96*a^3)

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 5881

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sinh[-a/b + x/b], x], x
, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -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}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}+\frac {1}{12} \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx-\frac {\int \frac {x \sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^3}+\frac {\int \frac {1}{\sqrt {\text {arccosh}(a x)}} \, dx}{6 a^2} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}+\frac {\text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{12 a^3}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{6 a^3} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{48 a^3}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^3}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{12 a^3} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{96 a^3}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{96 a^3}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{96 a^3}+\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{96 a^3}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{6 a^3} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^3}-\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{48 a^3}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{48 a^3}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{48 a^3}+\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{48 a^3} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{96 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.53 \[ \int x^2 \text {arccosh}(a x)^{3/2} \, dx=\frac {\sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-3 \text {arccosh}(a x)\right )+27 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-\text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (27 \Gamma \left (\frac {5}{2},\text {arccosh}(a x)\right )+\sqrt {3} \Gamma \left (\frac {5}{2},3 \text {arccosh}(a x)\right )\right )}{216 a^3 \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -3*ArcCosh[a*x]] + 27*Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -ArcCosh[a*x]] +
Sqrt[ArcCosh[a*x]]*(27*Gamma[5/2, ArcCosh[a*x]] + Sqrt[3]*Gamma[5/2, 3*ArcCosh[a*x]]))/(216*a^3*Sqrt[ArcCosh[a
*x]])

Maple [F]

\[\int x^{2} \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^2*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]

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

[In]

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

[Out]

Integral(x**2*acosh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*acosh(a*x)^(3/2), x)