\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^3 \, dx\) [704]

Optimal result

Integrand size = 22, antiderivative size = 136 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \arcsin (a x)}{a}-\frac {15 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \] Output:

1/8*c^3*(15*a*x+8)*(-a^2*x^2+1)^(1/2)/a^2/x-1/24*c^3*(-15*a*x+8)*(-a^2*x^2 
+1)^(3/2)/a^4/x^3+1/20*c^3*(-5*a*x+4)*(-a^2*x^2+1)^(5/2)/a^6/x^5+c^3*arcsi 
n(a*x)/a-15/8*c^3*arctanh((-a^2*x^2+1)^(1/2))/a
 

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.51 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\frac {7 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},a^2 x^2\right )}{x^5}-5 a^5 \left (1-a^2 x^2\right )^{7/2} \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},1-a^2 x^2\right )\right )}{35 a^6} \] Input:

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

Output:

(c^3*((7*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2])/x^5 - 5*a^5*(1 - a^ 
2*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 - a^2*x^2]))/(35*a^6)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6707, 6699, 537, 25, 537, 25, 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.

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

\(\Big \downarrow \) 6707

\(\displaystyle -\frac {c^3 \int \frac {e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6}dx}{a^6}\)

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 537

\(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \int -\frac {(4-5 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \int \frac {(4-5 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 537

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {(8-15 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \int \frac {(8-15 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 536

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (\int \frac {-8 x a^2-15 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(15 a x+8) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-8 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-15 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(15 a x+8) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-15 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (15 a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-\frac {15}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (15 a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (\frac {15 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (15 a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (15 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (15 a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\)

Input:

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

Output:

-((c^3*(-1/20*((4 - 5*a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/6*((8 - 15* 
a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (a^2*(-(((8 + 15*a*x)*Sqrt[1 - a^2*x^2])/x 
) - 8*a*ArcSin[a*x] + 15*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/a^6)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 0] && IntegerQ[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 6699

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]) && ILtQ[(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 = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\left (184 x^{6} a^{6}+135 a^{5} x^{5}-272 a^{4} x^{4}-165 a^{3} x^{3}+112 a^{2} x^{2}+30 a x -24\right ) c^{3}}{120 x^{5} \sqrt {-a^{2} x^{2}+1}\, a^{6}}+\frac {\left (-\frac {15 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}+\frac {a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a^{5} \sqrt {-a^{2} x^{2}+1}\right ) c^{3}}{a^{6}}\)	\(142\)
default	\(\frac {c^{3} \left (a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )+a^{5} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 x^{5}}-\frac {8 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 x^{3}}-2 a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-a^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{6}}\)	\(281\)

Input:

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

Output:

-1/120*(184*a^6*x^6+135*a^5*x^5-272*a^4*x^4-165*a^3*x^3+112*a^2*x^2+30*a*x 
-24)/x^5/(-a^2*x^2+1)^(1/2)*c^3/a^6+(-15/8*a^5*arctanh(1/(-a^2*x^2+1)^(1/2 
))+a^6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+a^5*(-a^2*x^2+ 
1)^(1/2))*c^3/a^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.13 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 225 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 120 \, a^{5} c^{3} x^{5} - {\left (120 \, a^{5} c^{3} x^{5} + 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x + 24 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}} \] Input:

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

Output:

-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 225*a^5*c 
^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 120*a^5*c^3*x^5 - (120*a^5*c^3*x^ 
5 + 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 - 88*a^2*c^3*x^2 - 30*a*c^3*x + 24*c 
^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.10 (sec) , antiderivative size = 692, normalized size of antiderivative = 5.09 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

c**3*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*as 
in(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 
 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**3*Piecewise((-I*a**2*x/sqr 
t(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x* 
*2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x** 
2 + 1)), True))/a**2 - 2*c**3*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sq 
rt(-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 + 2*c**3*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt 
(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/( 
a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + c**3*P 
iecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3* 
a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2)) 
), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 
1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqr 
t(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((2*I*a**4*sqrt(a**2*x* 
*2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 
 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a 
**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True)) 
/a**6
 

Maxima [F]

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

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

Output:

c^3*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - integrate((3*a^4*c^3*x^4 - 3* 
a^2*c^3*x^2 + c^3)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^7*x^7 + a^6*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (120) = 240\).

Time = 0.13 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.82 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {{\left (6 \, c^{3} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} + \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} + \frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} + \frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {15 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} + \frac {\frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{960 \, a^{4} {\left | a \right |}} \] Input:

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

Output:

-1/960*(6*c^3 - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) - 70*(sqrt( 
-a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4*x^2) + 240*(sqrt(-a^2*x^2 + 1)*abs(a) 
 + a)^3*c^3/(a^6*x^3) + 660*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^8*x^4 
))*a^10*x^5/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) + c^3*arcsin(a*x)*s 
gn(a)/abs(a) - 15/8*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 + 1/960*(660*(sqrt(-a^2*x^2 
+ 1)*abs(a) + a)*a^2*c^3/x + 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 
 - 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^2*x^3) - 15*(sqrt(-a^2*x^2 
+ 1)*abs(a) + a)^4*c^3/(a^4*x^4) + 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^3 
/(a^6*x^5))/(a^4*abs(a))
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^2\,x}+\frac {9\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^4\,x^3}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}+\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8\,a} \] Input:

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

Output:

(c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^3*atan((1 - a^2*x^2)^(1/2)*1 
i)*15i)/(8*a) + (c^3*(1 - a^2*x^2)^(1/2))/a + (23*c^3*(1 - a^2*x^2)^(1/2)) 
/(15*a^2*x) + (9*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) - (11*c^3*(1 - a^2*x 
^2)^(1/2))/(15*a^4*x^3) - (c^3*(1 - a^2*x^2)^(1/2))/(4*a^5*x^4) + (c^3*(1 
- a^2*x^2)^(1/2))/(5*a^6*x^5)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (120 \mathit {asin} \left (a x \right ) a^{5} x^{5}+120 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+184 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+135 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-88 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-30 \sqrt {-a^{2} x^{2}+1}\, a x +24 \sqrt {-a^{2} x^{2}+1}+225 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{5} x^{5}-150 a^{5} x^{5}\right )}{120 a^{6} x^{5}} \] Input:

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

Output:

(c**3*(120*asin(a*x)*a**5*x**5 + 120*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 18 
4*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 135*sqrt( - a**2*x**2 + 1)*a**3*x**3 
- 88*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 30*sqrt( - a**2*x**2 + 1)*a*x + 24 
*sqrt( - a**2*x**2 + 1) + 225*log(tan(asin(a*x)/2))*a**5*x**5 - 150*a**5*x 
**5))/(120*a**6*x**5)