\(\int \frac {1}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\) [102]

Optimal result

Integrand size = 14, antiderivative size = 128 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e^{-\frac {a}{b}} \left (-2 e^{a/b} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)-e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b d \sqrt {a+b \text {arccosh}(c+d x)}} \] Input:

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

Output:

(-2*E^(a/b)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) - E^((2*a)/b) 
*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[c + d*x]] + Sqrt[-( 
(a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)])/(b 
*d*E^(a/b)*Sqrt[a + b*ArcCosh[c + d*x]])
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6410, 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.

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

\(\Big \downarrow \) 6410

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

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

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

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]
 

rule 6410

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(acosh(c + d*x)*b + a)/(acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)* 
a*b + a**2),x)