\(\int \frac {e^{-3 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [593]

Optimal result

Integrand size = 24, antiderivative size = 199 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {(1-a x)^{5/2}}{a^2 \left (c-\frac {c}{a x}\right )^{5/2} x \sqrt {1+a x}}-\frac {2 (1-a x)^{5/2} \sqrt {1+a x}}{a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1-a x)^{5/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {(1-a x)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {2} a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}} \] Output:

(-a*x+1)^(5/2)/a^2/(c-c/a/x)^(5/2)/x/(a*x+1)^(1/2)-2*(-a*x+1)^(5/2)*(a*x+1 
)^(1/2)/a^3/(c-c/a/x)^(5/2)/x^2+(-a*x+1)^(5/2)*arcsinh(a^(1/2)*x^(1/2))/a^ 
(7/2)/(c-c/a/x)^(5/2)/x^(5/2)+1/2*(-a*x+1)^(5/2)*arctanh(2^(1/2)*a^(1/2)*x 
^(1/2)/(a*x+1)^(1/2))*2^(1/2)/a^(7/2)/(c-c/a/x)^(5/2)/x^(5/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.16 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-a x} \left (5 \left (2 \sqrt {a} \sqrt {x}-4 \sqrt {1+a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )+\sqrt {2+2 a x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )-4 a^{5/2} x^{5/2} \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-a x\right )\right )}{10 a^{3/2} c^2 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sqrt {1+a x}} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2)),x]
 

Output:

(Sqrt[1 - a*x]*(5*(2*Sqrt[a]*Sqrt[x] - 4*Sqrt[1 + a*x]*ArcSinh[Sqrt[a]*Sqr 
t[x]] + Sqrt[2 + 2*a*x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]) 
- 4*a^(5/2)*x^(5/2)*Sqrt[1 + a*x]*Hypergeometric2F1[3/2, 5/2, 7/2, -(a*x)] 
))/(10*a^(3/2)*c^2*Sqrt[c - c/(a*x)]*Sqrt[x]*Sqrt[1 + a*x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6684, 6679, 109, 27, 171, 27, 175, 63, 104, 219, 222}

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.

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

\(\Big \downarrow \) 6684

\(\displaystyle \frac {(1-a x)^{5/2} \int \frac {e^{-3 \text {arctanh}(a x)} x^{5/2}}{(1-a x)^{5/2}}dx}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\int \frac {\sqrt {x} (3-4 a x)}{2 (1-a x) \sqrt {a x+1}}dx}{a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\int \frac {\sqrt {x} (3-4 a x)}{(1-a x) \sqrt {a x+1}}dx}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {a (2-a x)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {2-a x}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx+\int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}+\int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}+2 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{3/2}}{a^2 \sqrt {a x+1}}-\frac {\frac {4 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\frac {2 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}}{a}}{2 a^2}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\)

Input:

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2)),x]
 

Output:

((1 - a*x)^(5/2)*(x^(3/2)/(a^2*Sqrt[1 + a*x]) - ((4*Sqrt[x]*Sqrt[1 + a*x]) 
/a - ((2*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a] + (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqr 
t[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[a])/a)/(2*a^2)))/((c - c/(a*x))^(5/2)*x 
^(5/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 6679

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Simp[c^p   Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] 
, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] 
|| GtQ[c, 0])
 

rule 6684

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p)   Int[u*(1 + c*(x/d))^p*(E^(n*Ar 
cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 
 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.40

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {2}\, \left (2 \sqrt {-x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +4 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x +\ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {-a^{2} x^{2}+1}}{4 a^{\frac {3}{2}} c^{3} \sqrt {-\frac {1}{a}}\, \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\)	\(279\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (-\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{2 a^{4} \sqrt {-2 c}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{a^{5} c \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{2}}\)	\(306\)

Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE 
)
 

Output:

-1/4*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)/c^3*2^(1/2)*(2*(-x*(a*x+1))^(1/2)*a^( 
5/2)*2^(1/2)*(-1/a)^(1/2)*x+a^2*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^ 
(1/2))*2^(1/2)*(-1/a)^(1/2)*x+4*(-x*(a*x+1))^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^ 
(1/2)+arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a*2^(1/2)*(-1/a)^(1 
/2)+a^(3/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-x*(a*x+1))^(1/2)*a-3*a*x-1)/(a*x- 
1))*x+ln((2*2^(1/2)*(-1/a)^(1/2)*(-x*(a*x+1))^(1/2)*a-3*a*x-1)/(a*x-1))*a^ 
(1/2))*(-a^2*x^2+1)^(1/2)/(-1/a)^(1/2)/(a*x+1)/(-x*(a*x+1))^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.47 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}\right ] \] Input:

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="fri 
cas")
 

Output:

[-1/8*(sqrt(2)*(a^2*x^2 - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 1 
3*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a* 
c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(a^2*x^2 - 1)* 
sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 
 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^2*x^2 + 2*a*x) 
*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - a*c^3), 1/4*(s 
qrt(2)*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c) 
*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + 2*(a^2*x^2 - 1)* 
sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2 
*a^2*c*x^2 - a*c*x - c)) - 4*(a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a* 
c*x - c)/(a*x)))/(a^3*c^3*x^2 - a*c^3)]
 

Sympy [F]

\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )^{3}}\, dx \] Input:

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(5/2),x)
 

Output:

Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 
1)**3), x)
 

Maxima [F]

\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="max 
ima")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^(5/2)), x)
 

Giac [F(-2)]

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

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="gia 
c")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (a\,x+1\right )}^3} \,d x \] Input:

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(5/2)*(a*x + 1)^3),x)
 

Output:

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(5/2)*(a*x + 1)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a x +\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )+2 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x +2 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )-2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x -4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{2 a \,c^{3} \left (a x +1\right )} \] Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x)
 

Output:

(sqrt(c)*(sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x + 1) 
)*a*x + sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x + 1)) 
+ 2*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a*x + 2*atan((sqrt(x 
)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) - 2*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a* 
i*x - 4*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(2*a*c**3*(a*x + 1))