3.2.59 \(\int \frac {1}{(a+b \text {arccosh}(c x))^{5/2}} \, dx\) [159]

3.2.59.1 Optimal result

Integrand size = 12, antiderivative size = 148 \[ \int \frac {1}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c} \]

output

-2/3*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/c+2/3 
*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/c/exp(a/b)-2/3*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(3/2)-4/3*x/b^2/(a+b*arcc 
osh(c*x))^(1/2)
 

3.2.59.2 Mathematica [F]

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

input

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

output

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

3.2.59.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6295, 6366, 6296, 25, 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.

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

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6366

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

\(\Big \downarrow \) 6296

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

(-2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (2* 
c*((-2*x)/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + ((2*I)*((I/2)*Sqrt[b]*E^(a/b)*S 
qrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Er 
fi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/E^(a/b)))/(b^2*c^2)))/(3*b)
 

3.2.59.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

3.2.59.4 Maple [F]

\[\int \frac {1}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}}d x\]

input

int(1/(a+b*arccosh(c*x))^(5/2),x)
 

output

int(1/(a+b*arccosh(c*x))^(5/2),x)
 

3.2.59.5 Fricas [F(-2)]

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

input

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

output

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

3.2.59.6 Sympy [F]

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

input

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

output

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

3.2.59.7 Maxima [F]

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

input

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

output

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

3.2.59.8 Giac [F]

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

input

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

output

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

3.2.59.9 Mupad [F(-1)]

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

input

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

output

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