\(\int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx\) [105]

Optimal result

Integrand size = 12, antiderivative size = 166 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3} \]

[Out]

-1/6*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3+1/6*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3-1/2*erf(3^(1/2)*arccosh(a*
x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/2*erfi(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/3*x^2*(a*x-1)^(1/2)
*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+8/3*x/a^2/arccosh(a*x)^(1/2)-4*x^3/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5886, 5951, 5887, 5556, 3389, 2211, 2235, 2236, 5881} \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}} \]

[In]

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

[Out]

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) + (8*x)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (4*x^3)/Sq
rt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(6*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(
2*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*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 5886

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] + (-Dist[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] + Dist[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 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 5951

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] - Dist[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, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}+12 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx-\frac {8 \int \frac {1}{\sqrt {\text {arccosh}(a x)}} \, dx}{3 a^2} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {8 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}+\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a^3}-\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a^3}+\frac {12 \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 )}{a^3} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}+\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}-\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}-\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}-\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a^3}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^3}+\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^3}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-\sqrt {\frac {-1+a x}{1+a x}} (1+a x)-3 e^{-3 \text {arccosh}(a x)} \text {arccosh}(a x)-e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-e^{\text {arccosh}(a x)} \text {arccosh}(a x)-3 e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)-3 \sqrt {3} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+3 \sqrt {3} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )-\sinh (3 \text {arccosh}(a x))}{6 a^3 \text {arccosh}(a x)^{3/2}} \]

[In]

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

[Out]

(-(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) - (3*ArcCosh[a*x])/E^(3*ArcCosh[a*x]) - ArcCosh[a*x]/E^ArcCosh[a*x] -
E^ArcCosh[a*x]*ArcCosh[a*x] - 3*E^(3*ArcCosh[a*x])*ArcCosh[a*x] - 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2,
-3*ArcCosh[a*x]] - (-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] + ArcCosh[a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x
]] + 3*Sqrt[3]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 3*ArcCosh[a*x]] - Sinh[3*ArcCosh[a*x]])/(6*a^3*ArcCosh[a*x]^(3/2)
)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/acosh(a*x)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/acosh(a*x)^(5/2), x)