\(\int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx\) [99]

Optimal result

Integrand size = 12, antiderivative size = 135 \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5885, 3388, 2211, 2235, 2236} \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(4*a^3) + (S
qrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(4*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(4*a^3) + (Sqrt[3*Pi]
*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(4*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 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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 5885

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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

Mathematica [A] (warning: unable to verify)

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.03 \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-\sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+\sqrt {3} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+2 \sinh (3 \text {arccosh}(a x))}{4 a^3 \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

-1/4*(2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) - Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -3*ArcCosh[a*x]] - Sqrt[
-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]] + Sqrt[3]*Sqrt[ArcCosh[
a*x]]*Gamma[1/2, 3*ArcCosh[a*x]] + 2*Sinh[3*ArcCosh[a*x]])/(a^3*Sqrt[ArcCosh[a*x]])

Maple [F]

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

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

[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]

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

[In]

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

[Out]

Integral(x**2/acosh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^(3/2), x)

Giac [F]

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

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

[In]

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

[Out]

int(x^2/acosh(a*x)^(3/2), x)