\(\int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx\) [333]

Optimal result

Integrand size = 19, antiderivative size = 92 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=\frac {a c^3 (4+a x) \sqrt {1-a^2 x^2}}{2 x}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+2 a^2 c^3 \arcsin (a x)-\frac {1}{2} a^2 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=\frac {c^3 \left (-2+8 a x+6 a^2 x^2-8 a^3 x^3-4 a^4 x^4+a^2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)-14 a^2 x^2 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-2 a^2 x^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{4 x^2 \sqrt {1-a^2 x^2}} \] Input:

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^3)/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]/Sqr 
t[2]] - 2*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(4*x^2*Sq 
rt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6678, 27, 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.

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

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]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {\left (2 a^{4} x^{4}+4 a^{3} x^{3}-3 a^{2} x^{2}-4 a x +1\right ) c^{3}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}-\left (\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {2 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{3}\)	\(106\)
default	\(-c^{3} \left (-a^{2} \sqrt {-a^{2} x^{2}+1}-\frac {2 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {2 a \sqrt {-a^{2} x^{2}+1}}{x}\right )\)	\(107\)
meijerg	\(\frac {a^{2} c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}+2 a^{2} c^{3} \arcsin \left (a x \right )+\frac {2 a \,c^{3} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a^{2} c^{3} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}\)	\(176\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=-\frac {8 \, a^{2} c^{3} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a^{2} c^{3} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, a^{2} c^{3} x^{2} - {\left (2 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x^{2}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/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*l 
og((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))/x^2
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.43 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=- a^{4} c^{3} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-a**4*c**3*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, T 
rue)) + 2*a**3*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*a*c**3*Piecewise((-I*sq 
rt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) 
 + 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))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=2 \, a^{2} c^{3} \arcsin \left (a x\right ) - \frac {1}{2} \, a^{2} c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a c^{3}}{x} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{2 \, x^{2}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (80) = 160\).

Time = 0.13 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.30 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=\frac {2 \, a^{3} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a^{3} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} + \frac {{\left (a^{3} c^{3} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{3}}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} + \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{3} {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=a^2\,c^3\,\sqrt {1-a^2\,x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {2\,a\,c^3\,\sqrt {1-a^2\,x^2}}{x}+\frac {2\,a^3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {a^2\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^3} \, dx=\frac {c^{3} \left (16 \mathit {asin} \left (a x \right ) a^{2} x^{2}+8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+16 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}-7 a^{2} x^{2}\right )}{8 x^{2}} \] Input:

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