\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a x})^3 \, dx\) [473]

Optimal result

Integrand size = 20, antiderivative size = 97 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {c^3 (4+a x) \sqrt {1-a^2 x^2}}{2 a^2 x}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {2 c^3 \arcsin (a x)}{a}+\frac {c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \] Output:

-1/2*c^3*(a*x+4)*(-a^2*x^2+1)^(1/2)/a^2/x+1/2*c^3*(-a^2*x^2+1)^(3/2)/a^3/x 
^2-2*c^3*arcsin(a*x)/a+1/2*c^3*arctanh((-a^2*x^2+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.64 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (2-8 a x-6 a^2 x^2+8 a^3 x^3+4 a^4 x^4-a^2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)+14 a^2 x^2 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+2 a^2 x^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{4 a^3 x^2 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(c^3*(2 - 8*a*x - 6*a^2*x^2 + 8*a^3*x^3 + 4*a^4*x^4 - a^2*x^2*Sqrt[1 - a^2 
*x^2]*ArcSin[a*x] + 14*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt 
[2]] + 2*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(4*a^3*x^2 
*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6681, 6678, 540, 27, 536, 538, 223, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6681

\(\displaystyle -\frac {c^3 \int \frac {e^{\text {arctanh}(a x)} (1-a x)^3}{x^3}dx}{a^3}\)

\(\Big \downarrow \) 6678

\(\displaystyle -\frac {c^3 \int \frac {(1-a x)^2 \sqrt {1-a^2 x^2}}{x^3}dx}{a^3}\)

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((c^3*(-1/2*(1 - a^2*x^2)^(3/2)/x^2 - (a*(-(((4 + a*x)*Sqrt[1 - a^2*x^2]) 
/x) - 4*a*ArcSin[a*x] + a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/a^3)
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 223

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[2*p]
 

rule 538

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

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 6678

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

rule 6681

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {c^{3} \left (-a^{2} \sqrt {-a^{2} x^{2}+1}-\frac {2 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {2 a \sqrt {-a^{2} x^{2}+1}}{x}\right )}{a^{3}}\)	\(109\)
risch	\(\frac {\left (2 a^{4} x^{4}+4 a^{3} x^{3}-3 a^{2} x^{2}-4 a x +1\right ) c^{3}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3}}+\frac {\left (\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {2 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{3}}{a^{3}}\)	\(111\)
meijerg	\(-\frac {c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}-\frac {2 c^{3} \arcsin \left (a x \right )}{a}-\frac {2 c^{3} \sqrt {-a^{2} x^{2}+1}}{a^{2} x}+\frac {c^{3} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 a \sqrt {\pi }}\)	\(178\)

Input:

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

Output:

c^3/a^3*(-a^2*(-a^2*x^2+1)^(1/2)-2*a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(- 
a^2*x^2+1)^(1/2))+1/2*(-a^2*x^2+1)^(1/2)/x^2+1/2*a^2*arctanh(1/(-a^2*x^2+1 
)^(1/2))-2*a*(-a^2*x^2+1)^(1/2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {8 \, a^{2} c^{3} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a^{2} c^{3} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, a^{2} c^{3} x^{2} - {\left (2 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}} \] Input:

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

Output:

1/2*(8*a^2*c^3*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a^2*c^3*x^2*lo 
g((sqrt(-a^2*x^2 + 1) - 1)/x) - 2*a^2*c^3*x^2 - (2*a^2*c^3*x^2 + 4*a*c^3*x 
 - c^3)*sqrt(-a^2*x^2 + 1))/(a^3*x^2)
 

Sympy [A] (verification not implemented)

Time = 3.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.31 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=a c^{3} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 2 c^{3} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + \frac {2 c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} \] Input:

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

Output:

a*c**3*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True) 
) - 2*c**3*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/ 
sqrt(-a**2), Ne(a**2, 0)), (x, True)) + 2*c**3*Piecewise((-I*sqrt(a**2*x** 
2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 - c** 
3*Piecewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1 
/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (I*a**2*asin( 
1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {2 \, c^{3} \arcsin \left (a x\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} + \frac {{\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{3}}{2 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x} \] Input:

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

Output:

-2*c^3*arcsin(a*x)/a - sqrt(-a^2*x^2 + 1)*c^3/a + 1/2*(a^2*log(2*sqrt(-a^2 
*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^3/a^3 - 2*sqrt(-a 
^2*x^2 + 1)*c^3/(a^2*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (85) = 170\).

Time = 0.15 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.13 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {{\left (c^{3} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} - \frac {2 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3} {\left | a \right |}}{a^{2} x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a^{4} x^{2}}}{8 \, a^{2}} \] Input:

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

Output:

-1/8*(c^3 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x))*a^4*x^2/((sqrt( 
-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)) - 2*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/ 
2*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) 
 - sqrt(-a^2*x^2 + 1)*c^3/a - 1/8*(8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3*a 
bs(a)/(a^2*x) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3*abs(a)/(a^4*x^2))/a^ 
2
 

Mupad [B] (verification not implemented)

Time = 15.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {2\,c^3\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {2\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \] Input:

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

Output:

(c^3*(1 - a^2*x^2)^(1/2))/(2*a^3*x^2) - (c^3*atan((1 - a^2*x^2)^(1/2)*1i)* 
1i)/(2*a) - (c^3*(1 - a^2*x^2)^(1/2))/a - (2*c^3*(1 - a^2*x^2)^(1/2))/(a^2 
*x) - (2*c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (-16 \mathit {asin} \left (a x \right ) a^{2} x^{2}-8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-16 \sqrt {-a^{2} x^{2}+1}\, a x +4 \sqrt {-a^{2} x^{2}+1}-4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}+7 a^{2} x^{2}\right )}{8 a^{3} x^{2}} \] Input:

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

Output:

(c**3*( - 16*asin(a*x)*a**2*x**2 - 8*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 16 
*sqrt( - a**2*x**2 + 1)*a*x + 4*sqrt( - a**2*x**2 + 1) - 4*log(tan(asin(a* 
x)/2))*a**2*x**2 + 7*a**2*x**2))/(8*a**3*x**2)