\(\int \frac {e^{-\text {arctanh}(a x)}}{(c-\frac {c}{a x})^{7/2}} \, dx\) [560]

Optimal result

Integrand size = 24, antiderivative size = 252 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {35 (1-a x)^{7/2} \sqrt {1+a x}}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}-\frac {5 (1-a x)^{7/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}+\frac {115 (1-a x)^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{16 \sqrt {2} a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}} \] Output:

1/4*(-a*x+1)^(3/2)*(a*x+1)^(1/2)/a^2/(c-c/a/x)^(7/2)/x-15/16*(-a*x+1)^(5/2 
)*(a*x+1)^(1/2)/a^3/(c-c/a/x)^(7/2)/x^2-35/16*(-a*x+1)^(7/2)*(a*x+1)^(1/2) 
/a^4/(c-c/a/x)^(7/2)/x^3-5*(-a*x+1)^(7/2)*arcsinh(a^(1/2)*x^(1/2))/a^(9/2) 
/(c-c/a/x)^(7/2)/x^(7/2)+115/32*(-a*x+1)^(7/2)*arctanh(2^(1/2)*a^(1/2)*x^( 
1/2)/(a*x+1)^(1/2))*2^(1/2)/a^(9/2)/(c-c/a/x)^(7/2)/x^(7/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1+a x} \left (35-55 a x+16 a^2 x^2\right )+160 (-1+a x)^2 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-115 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{32 a^{3/2} c^3 \sqrt {c-\frac {c}{a x}} \sqrt {x} (1-a x)^{3/2}} \] Input:

Integrate[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(7/2)),x]
 

Output:

(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(35 - 55*a*x + 16*a^2*x^2) + 160*(-1 + a* 
x)^2*ArcSinh[Sqrt[a]*Sqrt[x]] - 115*Sqrt[2]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]* 
Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(32*a^(3/2)*c^3*Sqrt[c - c/(a*x)]*Sqrt[x] 
*(1 - a*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6684, 6679, 109, 27, 166, 27, 171, 25, 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.

\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {(1-a x)^{7/2} \int \frac {e^{-\text {arctanh}(a x)} x^{7/2}}{(1-a x)^{7/2}}dx}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 6679

\(\displaystyle \frac {(1-a x)^{7/2} \int \frac {x^{7/2}}{(1-a x)^3 \sqrt {a x+1}}dx}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {\int \frac {5 x^{3/2} (2 a x+1)}{2 (1-a x)^2 \sqrt {a x+1}}dx}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \int \frac {x^{3/2} (2 a x+1)}{(1-a x)^2 \sqrt {a x+1}}dx}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {\int -\frac {a \sqrt {x} (14 a x+9)}{2 (1-a x) \sqrt {a x+1}}dx}{2 a^2}+\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\int \frac {\sqrt {x} (14 a x+9)}{(1-a x) \sqrt {a x+1}}dx}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {-\frac {\int -\frac {a (16 a x+7)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\int \frac {a (16 a x+7)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\int \frac {16 a x+7}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {23 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-16 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {23 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-32 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {46 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-32 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\frac {23 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-32 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{4 a^2 (1-a x)^2}-\frac {5 \left (\frac {3 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\frac {23 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {32 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}}{a}-\frac {14 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}\right )}{8 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\)

Input:

Int[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(7/2)),x]
 

Output:

((1 - a*x)^(7/2)*((x^(5/2)*Sqrt[1 + a*x])/(4*a^2*(1 - a*x)^2) - (5*((3*x^( 
3/2)*Sqrt[1 + a*x])/(2*a*(1 - a*x)) - ((-14*Sqrt[x]*Sqrt[1 + a*x])/a + ((- 
32*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a] + (23*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a 
]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[a])/a)/(4*a)))/(8*a^2)))/((c - c/(a*x))^(7 
/2)*x^(7/2))
 

Defintions of rubi rules used

rule 25

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

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 166

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

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 = 0.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.46

method	result	size
risch	\(-\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\left (\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{4} \sqrt {a^{2} c}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{4 a^{7} c \left (x -\frac {1}{a}\right )^{2}}+\frac {23 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{16 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {115 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{32 a^{5} \sqrt {-2 c}}\right ) a^{3} \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{3}}\)	\(367\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (-32 \sqrt {-x \left (a x +1\right )}\, a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+80 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, a^{3} x^{2}+110 \sqrt {-x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x -160 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x -115 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x^{2}-70 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+80 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+230 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x -115 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{64 a^{\frac {3}{2}} c^{4} \left (a x -1\right )^{3} \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\)	\(390\)

Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/a*(a*x+1)/(-(a*x+1)*a*c*x)^(1/2)/(c*(a*x-1)/a/x)^(1/2)*(c/(a*x-1)*a*x*( 
-a^2*x^2+1))^(1/2)/(-a^2*x^2+1)^(1/2)*(a*x-1)/c^3-(5/2/a^4/(a^2*c)^(1/2)*a 
rctan((a^2*c)^(1/2)*(x+1/2/a)/(-a^2*c*x^2-a*c*x)^(1/2))+1/4/a^7/c/(x-1/a)^ 
2*(-(x-1/a)^2*a^2*c-3*(x-1/a)*a*c-2*c)^(1/2)+23/16/a^6/c/(x-1/a)*(-(x-1/a) 
^2*a^2*c-3*(x-1/a)*a*c-2*c)^(1/2)-115/32/a^5/(-2*c)^(1/2)*ln((-4*c-3*(x-1/ 
a)*a*c+2*(-2*c)^(1/2)*(-(x-1/a)^2*a^2*c-3*(x-1/a)*a*c-2*c)^(1/2))/(x-1/a)) 
)*a^3/x/(c*(a*x-1)/a/x)^(1/2)*(c/(a*x-1)*a*x*(-a^2*x^2+1))^(1/2)/(-a^2*x^2 
+1)^(1/2)*(a*x-1)/c^3
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.38 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [-\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 8 \, {\left (16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{128 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),x, algorithm="frica 
s")
 

Output:

[-1/128*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(17*a 
^3*c*x^3 - 3*a^2*c*x^2 - 13*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)) + 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 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*(16*a^3*x^3 - 55*a^2*x^2 + 35*a*x)*sqrt(-a^ 
2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c 
^4*x - a*c^4), -1/64*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 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)) - 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 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*(16*a^3*x^3 - 55*a^2*x^2 + 35*a*x)*sqrt(-a^2*x^2 + 
 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - 
a*c^4)]
 

Sympy [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}\, dx \] Input:

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(7/2),x)
 

Output:

Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x)))**(7/2)*(a*x + 1)) 
, x)
 

Maxima [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(7/2)), x)
 

Giac [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:

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

Output:

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x+1\right )} \,d x \] Input:

int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(7/2)*(a*x + 1)),x)
 

Output:

int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(7/2)*(a*x + 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-115 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a^{2} x^{2}+230 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a x -115 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )+160 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{2} x^{2}-320 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x +160 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )+32 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}-110 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +70 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{32 a \,c^{4} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),x)
 

Output:

(sqrt(c)*( - 115*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a 
*x + 1))*a**2*x**2 + 230*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt( 
2)*i)/(a*x + 1))*a*x - 115*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqr 
t(2)*i)/(a*x + 1)) + 160*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) 
*a**2*x**2 - 320*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a*x + 1 
60*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) + 32*sqrt(x)*sqrt(a)* 
sqrt(a*x + 1)*a**2*i*x**2 - 110*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x + 70*s 
qrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(32*a*c**4*(a**2*x**2 - 2*a*x + 1))