\(\int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx\) [107]

Optimal result

Integrand size = 8, antiderivative size = 89 \[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5880, 5951, 5881, 3389, 2211, 2235, 2236} \[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{3 a}-\frac {4 x}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}} \]

[In]

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

[Out]

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

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 5880

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

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

Mathematica [A] (warning: unable to verify)

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {2 \left (-\sqrt {\frac {-1+a x}{1+a x}} (1+a x)-e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-e^{\text {arccosh}(a x)} \text {arccosh}(a x)-(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )}{3 a \text {arccosh}(a x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {2 \left (2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x +\operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\operatorname {arccosh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\right )}{3 \sqrt {\pi }\, a \operatorname {arccosh}\left (a x \right )^{2}}\)	\(84\)

[In]

int(1/arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/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]

\[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {1}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/acosh(a*x)**(5/2),x)

[Out]

Integral(acosh(a*x)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

int(1/acosh(a*x)^(5/2),x)

[Out]

int(1/acosh(a*x)^(5/2), x)