\(\int \frac {e^{3 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^2} \, dx\) [691]

Optimal result

Integrand size = 22, antiderivative size = 124 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {12 (1+a x)}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (5+4 a x)}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {3 \arcsin (a x)}{a c^2} \] Output:

1/5*(a*x+1)^3/a/c^2/(-a^2*x^2+1)^(5/2)-12/5*(a*x+1)/a/c^2/(-a^2*x^2+1)^(3/ 
2)+6/5*(4*a*x+5)/a/c^2/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a/c^2-3*arcsi 
n(a*x)/a/c^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {24-33 a x-18 a^2 x^2+34 a^3 x^3-5 a^4 x^4-15 (-1+a x)^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a c^2 (-1+a x)^2 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(24 - 33*a*x - 18*a^2*x^2 + 34*a^3*x^3 - 5*a^4*x^4 - 15*(-1 + a*x)^2*Sqrt[ 
1 - a^2*x^2]*ArcSin[a*x])/(5*a*c^2*(-1 + a*x)^2*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6707, 6698, 529, 2166, 27, 2166, 27, 455, 223}

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.

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

\(\Big \downarrow \) 6707

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {a^4 \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {3 (a x+1) \left (\frac {5 x^2}{a^2}+\frac {10 x}{a^3}+\frac {9}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^4 \left (\frac {1}{5} \left (\int \frac {(a x+1) \left (\frac {5 x^2}{a^2}+\frac {10 x}{a^3}+\frac {9}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^2}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {a^4 \left (\frac {1}{5} \left (-\int \frac {5 (a x+3)}{a^4 \sqrt {1-a^2 x^2}}dx-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{a^5 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^4 \left (\frac {1}{5} \left (-\frac {5 \int \frac {a x+3}{\sqrt {1-a^2 x^2}}dx}{a^4}-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{a^5 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {a^4 \left (\frac {1}{5} \left (-\frac {5 \left (3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^4}-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{a^5 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {a^4 \left (\frac {(a x+1)^3}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (-\frac {6 (a x+1)^2}{a^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{a^5 \sqrt {1-a^2 x^2}}-\frac {5 \left (\frac {3 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^4}\right )\right )}{c^2}\)

Input:

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

Output:

(a^4*((1 + a*x)^3/(5*a^5*(1 - a^2*x^2)^(5/2)) + ((-6*(1 + a*x)^2)/(a^5*(1 
- a^2*x^2)^(3/2)) + (24*(1 + a*x))/(a^5*Sqrt[1 - a^2*x^2]) - (5*(-(Sqrt[1 
- a^2*x^2]/a) + (3*ArcSin[a*x])/a))/a^4)/5))/c^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 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 455

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

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 6698

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

rule 6707

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.56

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {\left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}+\frac {6 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {24 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{6} \left (x -\frac {1}{a}\right )}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}\right ) a^{4}}{c^{2}}\)	\(193\)
default	\(\frac {a^{4} \left (\frac {-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}}{a}+\frac {7 x}{a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3}{a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {3 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{5}}+\frac {5}{a^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {\frac {2}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {6 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{6}}\right )}{c^{2}}\)	\(374\)

Input:

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

Output:

-1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^2-(1/5/a^8/(x-1/a)^3*(-(x-1/a)^2*a^2 
-2*a*(x-1/a))^(1/2)+6/5/a^7/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+2 
4/5/a^6/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+3/a^4/(a^2)^(1/2)*arcta 
n((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))*a^4/c^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right )} \sqrt {-a^{2} x^{2} + 1} - 24}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \] Input:

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

Output:

1/5*(24*a^3*x^3 - 72*a^2*x^2 + 72*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 
1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (5*a^3*x^3 - 39*a^2*x^2 + 57*a 
*x - 24)*sqrt(-a^2*x^2 + 1) - 24)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2 
*x - a*c^2)
 

Sympy [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \left (\int \frac {x^{4}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \] Input:

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

Output:

a**4*(Integral(x**4/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a 
**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + I 
ntegral(a*x**5/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x 
**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.45 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} - \frac {2 \, {\left (\frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 19\right )}}{5 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \] Input:

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

Output:

-3*arcsin(a*x)*sgn(a)/(c^2*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c^2) - 2/5*(80* 
(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 120*(sqrt(-a^2*x^2 + 1)*abs(a) + 
 a)^2/(a^4*x^2) + 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 15*(sqr 
t(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 19)/(c^2*((sqrt(-a^2*x^2 + 1)*ab 
s(a) + a)/(a^2*x) - 1)^5*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.19 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,c^2\,x^2-2\,a^6\,c^2\,x+a^5\,c^2\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}+\frac {24\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}+a^2\,c^2\,x^3\,\sqrt {-a^2}-3\,a\,c^2\,x^2\,\sqrt {-a^2}\right )} \] Input:

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

Output:

(2*a^4*(1 - a^2*x^2)^(1/2))/(15*(a^5*c^2 - 2*a^6*c^2*x + a^7*c^2*x^2)) - ( 
3*asinh(x*(-a^2)^(1/2)))/(c^2*(-a^2)^(1/2)) - (4*a*(1 - a^2*x^2)^(1/2))/(3 
*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) + (1 - a^2*x^2)^(1/2)/(a*c^2) + (2 
4*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) - (c^2*(-a^2)^( 
1/2))/a)) + (1 - a^2*x^2)^(1/2)/(5*(-a^2)^(1/2)*(3*c^2*x*(-a^2)^(1/2) - (c 
^2*(-a^2)^(1/2))/a + a^2*c^2*x^3*(-a^2)^(1/2) - 3*a*c^2*x^2*(-a^2)^(1/2)))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.16 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {-15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+30 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-15 \mathit {asin} \left (a x \right ) a^{3} x^{3}+45 \mathit {asin} \left (a x \right ) a^{2} x^{2}-45 \mathit {asin} \left (a x \right ) a x +15 \mathit {asin} \left (a x \right )+5 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-21 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+21 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}-5 a^{4} x^{4}+52 a^{3} x^{3}-72 a^{2} x^{2}+21 a x +6}{5 a \,c^{2} \left (\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 \sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:

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

Output:

( - 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 30*sqrt( - a**2*x**2 + 
 1)*asin(a*x)*a*x - 15*sqrt( - a**2*x**2 + 1)*asin(a*x) - 15*asin(a*x)*a** 
3*x**3 + 45*asin(a*x)*a**2*x**2 - 45*asin(a*x)*a*x + 15*asin(a*x) + 5*sqrt 
( - a**2*x**2 + 1)*a**3*x**3 - 21*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 21*sq 
rt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x**2 + 1) - 5*a**4*x**4 + 52*a** 
3*x**3 - 72*a**2*x**2 + 21*a*x + 6)/(5*a*c**2*(sqrt( - a**2*x**2 + 1)*a**2 
*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 1) + a**3*x**3 
- 3*a**2*x**2 + 3*a*x - 1))