\(\int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [175]

Optimal result

Integrand size = 12, antiderivative size = 120 \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c} \] Output:

-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)+exp(a/b)*Pi^(1 
/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c+Pi^(1/2)*erfi((a+b*arc 
cosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c/exp(a/b)
 

Mathematica [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, 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.

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

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

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

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 [F]

\[\int \frac {1}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(a+b*arccosh(c*x))^(3/2),x)
 

Output:

int(1/(a+b*arccosh(c*x))^(3/2),x)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*acosh(c*x) - acosh(c 
*x)*int((sqrt(acosh(c*x)*b + a)*acosh(c*x)*x**2)/(acosh(c*x)**2*b**2*c**2* 
x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b 
+ a**2*c**2*x**2 - a**2),x)*b**2*c**3 + acosh(c*x)*int((sqrt(acosh(c*x)*b 
+ a)*acosh(c*x))/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*ac 
osh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b**2 
*c - 2*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)* 
acosh(c*x)*x)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh 
(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b*c** 
2 - 2*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*a 
cosh(c*x)**2*x)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*aco 
sh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b**2* 
c**2 - int((sqrt(acosh(c*x)*b + a)*acosh(c*x)*x**2)/(acosh(c*x)**2*b**2*c* 
*2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a 
*b + a**2*c**2*x**2 - a**2),x)*a*b*c**3 + int((sqrt(acosh(c*x)*b + a)*acos 
h(c*x))/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)* 
a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b*c - 2*int 
((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*acosh(c*x)*x)/(acosh( 
c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 
 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a**2*c**2 - 2*int((sqrt(c...