3.5.14 \(\int \frac {(c-a^2 c x^2)^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx\) [414]

3.5.14.1 Optimal result

Integrand size = 24, antiderivative size = 329 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \]

output

-2/3*(-a^2*c*x^2+c)^(3/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2) 
-2/3*c*erf(2*arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^( 
1/2)/(a*x+1)^(1/2)-2/3*c*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c 
)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+2/3*c*erf(2^(1/2)*arccosh(a*x)^(1/2) 
)*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+2/3* 
c*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a 
/(a*x-1)^(1/2)/(a*x+1)^(1/2)-16/3*c*x*(-a*x+1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2 
)/arccosh(a*x)^(1/2)
 

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

Time = 0.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {c e^{-4 \text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \left (-1-14 e^{4 \text {arccosh}(a x)}-e^{8 \text {arccosh}(a x)}+16 a^2 e^{4 \text {arccosh}(a x)} x^2+8 \text {arccosh}(a x)-8 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)+64 a e^{4 \text {arccosh}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+64 a^2 e^{4 \text {arccosh}(a x)} x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)-16 e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )+16 \sqrt {2} e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+16 \sqrt {2} e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-16 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )}{24 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^{3/2}} \]

input

Integrate[(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(5/2),x]
 

output

-1/24*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a* 
x]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2 + 8*ArcCosh[a*x] - 8*E^(8*ArcCosh[a*x] 
)*ArcCosh[a*x] + 64*a*E^(4*ArcCosh[a*x])*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcC 
osh[a*x] + 64*a^2*E^(4*ArcCosh[a*x])*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCos 
h[a*x] - 16*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -4*ArcCosh 
[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -2 
*ArcCosh[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/ 
2, 2*ArcCosh[a*x]] - 16*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 4 
*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x 
)*ArcCosh[a*x]^(3/2))
 

3.5.14.3 Rubi [A] (verified)

Time = 2.20 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.76, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6319, 25, 6327, 6357, 6322, 3042, 25, 3793, 2009, 6368, 5971, 2009}

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.

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

\(\Big \downarrow \) 6319

\(\displaystyle -\frac {8 a c \sqrt {c-a^2 c x^2} \int -\frac {x (1-a x) (a x+1)}{\text {arccosh}(a x)^{3/2}}dx}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \int \frac {x (1-a x) (a x+1)}{\text {arccosh}(a x)^{3/2}}dx}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \int \frac {x \left (1-a^2 x^2\right )}{\text {arccosh}(a x)^{3/2}}dx}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6357

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 \int \frac {\sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx}{a}+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6322

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (\frac {2 \int \frac {(a x-1) (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 a c \sqrt {c-a^2 c x^2} \left (\frac {2 \int -\frac {\sin (i \text {arccosh}(a x))^2}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 a c \sqrt {c-a^2 c x^2} \left (-\frac {2 \int \frac {\sin (i \text {arccosh}(a x))^2}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-\frac {2 \int \left (\frac {1}{2 \sqrt {\text {arccosh}(a x)}}-\frac {\cosh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^2}-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-8 a \int \frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{\sqrt {\text {arccosh}(a x)}}dx+\frac {2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-\frac {8 \int \frac {a^2 x^2 (a x-1) (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}+\frac {2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-\frac {8 \int \left (\frac {\cosh (4 \text {arccosh}(a x))}{8 \sqrt {\text {arccosh}(a x)}}-\frac {1}{8 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^2}+\frac {2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 a c \sqrt {c-a^2 c x^2} \left (-\frac {8 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\sqrt {\text {arccosh}(a x)}\right )}{a^2}+\frac {2 x (a x-1)^{3/2} (a x+1)^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}\)

input

Int[(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(5/2),x]
 

output

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcCosh[a*x]^ 
(3/2)) + (8*a*c*Sqrt[c - a^2*c*x^2]*((2*x*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2) 
)/(a*Sqrt[ArcCosh[a*x]]) - (8*(-1/4*Sqrt[ArcCosh[a*x]] + (Sqrt[Pi]*Erf[2*S 
qrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32))/a^2 + 
(2*(-Sqrt[ArcCosh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4 + 
 (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4))/a^2))/(3*Sqrt[-1 + a*x] 
*Sqrt[1 + a*x])
 

3.5.14.3.1 Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6319

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A 
rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si 
mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[x*(1 + c*x)^(p - 1/2)*(- 
1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, 
d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
 

rule 6322

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d1 + e1*x)^p/ 
(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[Int[x^n*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[2*p, 0]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x 
)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

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]
 

3.5.14.4 Maple [F]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]

input

int((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x)
 

output

int((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x)
 

3.5.14.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x, algorithm="fricas")
 

output

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

3.5.14.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Timed out} \]

input

integrate((-a**2*c*x**2+c)**(3/2)/acosh(a*x)**(5/2),x)
 

output

Timed out
 

3.5.14.7 Maxima [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x, algorithm="maxima")
 

output

integrate((-a^2*c*x^2 + c)^(3/2)/arccosh(a*x)^(5/2), x)
 

3.5.14.8 Giac [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(5/2),x, algorithm="giac")
 

output

integrate((-a^2*c*x^2 + c)^(3/2)/arccosh(a*x)^(5/2), x)
 

3.5.14.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]

input

int((c - a^2*c*x^2)^(3/2)/acosh(a*x)^(5/2),x)
 

output

int((c - a^2*c*x^2)^(3/2)/acosh(a*x)^(5/2), x)