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

Optimal result

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

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

Mathematica [A] (warning: unable to verify)

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

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

Output:

(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] - (-ArcCosh[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))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6295, 6366, 6296, 3042, 26, 3789, 2611, 2633, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6295

\(\displaystyle \frac {2}{3} a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6366

\(\displaystyle \frac {2}{3} a \left (\frac {2 \int \frac {1}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6296

\(\displaystyle \frac {2}{3} a \left (\frac {2 \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int -\frac {i \sin (i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \int \frac {\sin (i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (i \int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-i \int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2633

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 2634

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

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

rule 6295

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] - Simp[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 6296

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

rule 6366

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94

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

Input:

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

Output:

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

Integral(acosh(a*x)**(-5/2), x)
 

Maxima [F]

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

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

Output:

integrate(arccosh(a*x)^(-5/2), x)
 

Giac [F]

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

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

Output:

integrate(arccosh(a*x)^(-5/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {\frac {2 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}}{3}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}}{3}}{\mathit {acosh} \left (a x \right )^{2} a} \] Input:

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

Output:

(2*(acosh(a*x)**2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(ac 
osh(a*x)**2*a**2*x**2 - acosh(a*x)**2),x)*a**2 - sqrt(a*x + 1)*sqrt(a*x - 
1)*sqrt(acosh(a*x))))/(3*acosh(a*x)**2*a)