\(\int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx\) [101]

Optimal result

Integrand size = 8, antiderivative size = 68 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{a} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )}{a \sqrt {\text {arccosh}(a x)}} \] Input:

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

Output:

(-2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + Sqrt[-ArcCosh[a*x]]*Gamma[1/2, 
-ArcCosh[a*x]] - Sqrt[ArcCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]])/(a*Sqrt[ArcC 
osh[a*x]])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6295, 6368, 3042, 3788, 26, 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.

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

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6368

\(\displaystyle \frac {2 \int \frac {a x}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (2*((Sqrt[Pi]*E 
rf[Sqrt[ArcCosh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/2))/a
 

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 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[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 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 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {-2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\operatorname {arccosh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{\sqrt {\pi }\, a \,\operatorname {arccosh}\left (a x \right )}\)	\(66\)

Input:

int(1/arccosh(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/arccosh(a*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(1/acosh(a*x)**(3/2),x)
 

Output:

Integral(acosh(a*x)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/arccosh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/arccosh(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(arccosh(a*x)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(1/acosh(a*x)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(1/acosh(a*x)^(3/2), x)
 

Reduce [F]

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

int(1/acosh(a*x)^(3/2),x)
 

Output:

(2*(acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(acosh 
(a*x)*a**2*x**2 - acosh(a*x)),x)*a**2 - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(a 
cosh(a*x))))/(acosh(a*x)*a)