3.2.13 \(\int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx\) [113]

3.2.13.1 Optimal result

Integrand size = 8, antiderivative size = 122 \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}-\frac {4 x}{15 \text {arccosh}(a x)^{3/2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{15 a} \]

output

-4/15*x/arccosh(a*x)^(3/2)+4/15*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a+4/15*er 
fi(arccosh(a*x)^(1/2))*Pi^(1/2)/a-2/5*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccos 
h(a*x)^(5/2)-8/15*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)
 

3.2.13.2 Mathematica [A] (warning: unable to verify)

Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 e^{-\text {arccosh}(a x)} \left (3 e^{\text {arccosh}(a x)} \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x)+e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)-2 \text {arccosh}(a x)^2+2 e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2-2 e^{\text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+2 e^{\text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )}{15 a \text {arccosh}(a x)^{5/2}} \]

input

Integrate[ArcCosh[a*x]^(-7/2),x]
 

output

(-2*(3*E^ArcCosh[a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x] 
+ E^(2*ArcCosh[a*x])*ArcCosh[a*x] - 2*ArcCosh[a*x]^2 + 2*E^(2*ArcCosh[a*x] 
)*ArcCosh[a*x]^2 - 2*E^ArcCosh[a*x]*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -ArcC 
osh[a*x]] + 2*E^ArcCosh[a*x]*ArcCosh[a*x]^(5/2)*Gamma[1/2, ArcCosh[a*x]])) 
/(15*a*E^ArcCosh[a*x]*ArcCosh[a*x]^(5/2))
 

3.2.13.3 Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6295, 6366, 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.

\(\displaystyle \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6366

\(\displaystyle \frac {2}{5} a \left (\frac {2 \int \frac {1}{\text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 6295

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (\frac {2 \int \frac {a x}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {2}{5} a \left (-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a}\right )}{3 a}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {2}{5} a \left (-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}\right )}{3 a}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)+\frac {1}{2} \int \frac {e^{\text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (\frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\int e^{\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (\frac {2 \left (\int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {2}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

input

Int[ArcCosh[a*x]^(-7/2),x]
 

output

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) + (2*a*((-2*x)/ 
(3*a*ArcCosh[a*x]^(3/2)) + (2*((-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[A 
rcCosh[a*x]]) + (2*((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/2 + (Sqrt[Pi]*Erfi[ 
Sqrt[ArcCosh[a*x]]])/2))/a))/(3*a)))/5
 

3.2.13.3.1 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 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]
 

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]
 

3.2.13.4 Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {-\frac {8 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}{15}+\frac {4 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{15}+\frac {4 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{15}-\frac {4 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x}{15}-\frac {2 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}}{5}}{\sqrt {\pi }\, a \operatorname {arccosh}\left (a x \right )^{3}}\)	\(111\)

input

int(1/arccosh(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

output

2/15*(-4*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Pi^(1/2)*arccosh(a*x)^(5/2)+2*arccosh 
(a*x)^3*Pi*erf(arccosh(a*x)^(1/2))+2*arccosh(a*x)^3*Pi*erfi(arccosh(a*x)^( 
1/2))-2*arccosh(a*x)^(3/2)*Pi^(1/2)*a*x-3*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)^3
 

3.2.13.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(1/arccosh(a*x)^(7/2),x, algorithm="fricas")
 

output

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

3.2.13.6 Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \]

input

integrate(1/acosh(a*x)**(7/2),x)
 

output

Timed out
 

3.2.13.7 Maxima [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

input

integrate(1/arccosh(a*x)^(7/2),x, algorithm="maxima")
 

output

integrate(arccosh(a*x)^(-7/2), x)
 

3.2.13.8 Giac [F]

\[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

input

integrate(1/arccosh(a*x)^(7/2),x, algorithm="giac")
 

output

integrate(arccosh(a*x)^(-7/2), x)
 

3.2.13.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \]

input

int(1/acosh(a*x)^(7/2),x)
 

output

int(1/acosh(a*x)^(7/2), x)