\(\int e^{3 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\) [671]

Optimal result

Integrand size = 22, antiderivative size = 246 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {15 c^4 \sqrt {1-a^2 x^2}}{16 a}-\frac {3 c^4 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{a^4 x^3}+\frac {5 c^4 \left (1-a^2 x^2\right )^{3/2}}{16 a^3 x^2}-\frac {3 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{8 a^5 x^4}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \arcsin (a x)}{a}-\frac {15 c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{16 a} \] Output:

15/16*c^4*(-a^2*x^2+1)^(1/2)/a-3*c^4*(-a^2*x^2+1)^(1/2)/a^2/x+c^4*(-a^2*x^ 
2+1)^(3/2)/a^4/x^3+5/16*c^4*(-a^2*x^2+1)^(3/2)/a^3/x^2-3/5*c^4*(-a^2*x^2+1 
)^(5/2)/a^6/x^5-1/8*c^4*(-a^2*x^2+1)^(5/2)/a^5/x^4-1/7*c^4*(-a^2*x^2+1)^(7 
/2)/a^8/x^7-1/2*c^4*(-a^2*x^2+1)^(7/2)/a^7/x^6-3*c^4*arcsin(a*x)/a-15/16*c 
^4*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 = 0.20 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (-336 a^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},a^2 x^2\right )-\frac {5 \left (16+56 a x-64 a^2 x^2-238 a^3 x^3+96 a^4 x^4+413 a^5 x^5-64 a^6 x^6-231 a^7 x^7+16 a^8 x^8-105 a^7 x^7 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+16 a^7 x^7 \left (-1+a^2 x^2\right )^4 \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},1-a^2 x^2\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{560 a^8 x^7} \] Input:

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]
 

Output:

(c^4*(-336*a^2*x^2*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2] - (5*(16 + 
 56*a*x - 64*a^2*x^2 - 238*a^3*x^3 + 96*a^4*x^4 + 413*a^5*x^5 - 64*a^6*x^6 
 - 231*a^7*x^7 + 16*a^8*x^8 - 105*a^7*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 
 - a^2*x^2]] + 16*a^7*x^7*(-1 + a^2*x^2)^4*Hypergeometric2F1[3, 7/2, 9/2, 
1 - a^2*x^2]))/Sqrt[1 - a^2*x^2]))/(560*a^8*x^7)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {6707, 6698, 540, 27, 2338, 27, 537, 25, 537, 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.

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

\(\Big \downarrow \) 6707

\(\displaystyle \frac {c^4 \int \frac {e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^4}{x^8}dx}{a^8}\)

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

\(\displaystyle \frac {c^4 \left (-\frac {1}{6} \int -\frac {3 a^2 (a x+6) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \int \frac {(a x+6) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (\frac {1}{4} a^2 \int -\frac {(5 a x+24) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \int \frac {(5 a x+24) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (5 a x+16) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(5 a x+16) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 536

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {5 a-16 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(16-5 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-16 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(16-5 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (16-5 a x)}{x}-16 a \arcsin (a x)\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {5}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (16-5 a x)}{x}-16 a \arcsin (a x)\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (16-5 a x)}{x}-16 a \arcsin (a x)\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {c^4 \left (\frac {1}{2} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (16-5 a x)}{x}-16 a \arcsin (a x)\right )-\frac {(5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{7 x^7}-\frac {a \left (1-a^2 x^2\right )^{7/2}}{2 x^6}\right )}{a^8}\)

Input:

Int[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]
 

Output:

(c^4*(-1/7*(1 - a^2*x^2)^(7/2)/x^7 - (a*(1 - a^2*x^2)^(7/2))/(2*x^6) + (a^ 
2*(-1/20*((24 + 5*a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/2*((16 + 5*a*x) 
*(1 - a^2*x^2)^(3/2))/x^3 - (3*a^2*(-(((16 - 5*a*x)*Sqrt[1 - a^2*x^2])/x) 
- 16*a*ArcSin[a*x] - 5*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/2))/a^8
 

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 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 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 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

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 = 0.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {\left (560 a^{9} x^{9}-2496 a^{8} x^{8}-1085 a^{7} x^{7}+3488 x^{6} a^{6}+1295 a^{5} x^{5}-1088 a^{4} x^{4}-1050 a^{3} x^{3}+16 a^{2} x^{2}+280 a x +80\right ) c^{4}}{560 x^{7} \sqrt {-a^{2} x^{2}+1}\, a^{8}}+\frac {\left (-\frac {15 a^{7} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {3 a^{8} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{4}}{a^{8}}\)	\(151\)
default	\(\frac {c^{4} \left (-\frac {1}{7 x^{7} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{2} \left (-\frac {1}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {6 a^{2} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )}{5}\right )}{7}+a^{11} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {11 a^{8} x}{\sqrt {-a^{2} x^{2}+1}}+3 a \left (-\frac {1}{6 x^{6} \sqrt {-a^{2} x^{2}+1}}+\frac {7 a^{2} \left (-\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )}{6}\right )-11 a^{3} \left (-\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )-6 a^{4} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )+14 a^{5} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+14 a^{6} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {a^{7}}{\sqrt {-a^{2} x^{2}+1}}+3 a^{10} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-6 a^{7} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\right )}{a^{8}}\)	\(616\)
meijerg	\(-\frac {11 c^{4} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {14 c^{4} \left (-2 a^{2} x^{2}+1\right )}{a^{2} x \sqrt {-a^{2} x^{2}+1}}+\frac {2 c^{4} \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {c^{4} \left (-16 x^{6} a^{6}+8 a^{4} x^{4}+2 a^{2} x^{2}+1\right )}{5 a^{6} x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4} \left (-128 a^{8} x^{8}+64 x^{6} a^{6}+16 a^{4} x^{4}+8 a^{2} x^{2}+5\right ) \sqrt {-a^{2} x^{2}+1}}{7 a^{8} x^{7} \left (-5 a^{2} x^{2}+5\right )}+\frac {c^{4} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{4} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {6 c^{4} \left (\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{a \sqrt {\pi }}-\frac {14 c^{4} \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{a \sqrt {\pi }}-\frac {11 c^{4} \left (-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}\right )}{a \sqrt {\pi }}-\frac {3 c^{4} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{4} \left (\frac {\sqrt {\pi }}{6 x^{6} a^{6}}+\frac {3 \sqrt {\pi }}{8 x^{4} a^{4}}+\frac {15 \sqrt {\pi }}{16 x^{2} a^{2}}-\frac {35 \left (\frac {319}{210}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{32}-\frac {\sqrt {\pi }\, \left (-1276 x^{6} a^{6}+720 a^{4} x^{4}+288 a^{2} x^{2}+128\right )}{768 a^{6} x^{6}}+\frac {\sqrt {\pi }\, \left (-1680 x^{6} a^{6}+560 a^{4} x^{4}+224 a^{2} x^{2}+128\right )}{768 a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}}+\frac {35 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{16}\right )}{a \sqrt {\pi }}\)	\(865\)

Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/560*(560*a^9*x^9-2496*a^8*x^8-1085*a^7*x^7+3488*a^6*x^6+1295*a^5*x^5-10 
88*a^4*x^4-1050*a^3*x^3+16*a^2*x^2+280*a*x+80)/x^7/(-a^2*x^2+1)^(1/2)*c^4/ 
a^8+(-15/16*a^7*arctanh(1/(-a^2*x^2+1)^(1/2))-3*a^8/(a^2)^(1/2)*arctan((a^ 
2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))*c^4/a^8
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.71 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 525 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 560 \, a^{7} c^{4} x^{7} + {\left (560 \, a^{7} c^{4} x^{7} - 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} + 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} - 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="frica 
s")
 

Output:

1/560*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 525*a^7*c 
^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 560*a^7*c^4*x^7 + (560*a^7*c^4*x^ 
7 - 2496*a^6*c^4*x^6 - 525*a^5*c^4*x^5 + 992*a^4*c^4*x^4 + 770*a^3*c^4*x^3 
 - 96*a^2*c^4*x^2 - 280*a*c^4*x - 80*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)
 

Sympy [A] (verification not implemented)

Time = 19.04 (sec) , antiderivative size = 932, normalized size of antiderivative = 3.79 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**4,x)
 

Output:

-a*c**4*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True 
)) - 3*c**4*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)) + 8*c**4*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 + 6* 
c**4*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*as 
in(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 - 6*c**4*Pi 
ecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x* 
*3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2 
*x**2 + 1)/(3*x**3), True))/a**4 - 8*c**4*Piecewise((-3*a**4*acosh(1/(a*x) 
)/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - 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), (3 
*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8* 
x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True 
))/a**5 + 3*c**4*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*x*sqrt( 
-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x 
**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/A 
bs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*sqrt(1 - 1 
/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5 
*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True)...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (214) = 428\).

Time = 0.12 (sec) , antiderivative size = 745, normalized size of antiderivative = 3.03 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx =\text {Too large to display} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="maxim 
a")
 

Output:

-a^3*c^4*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a 
^2*c^4*(x/(sqrt(-a^2*x^2 + 1)*a^2) - arcsin(a*x)/a^3) - 11*c^4*x/sqrt(-a^2 
*x^2 + 1) - 6*c^4*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x) 
+ 2/abs(x)))/a + 14*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x) 
)*c^4/a^2 - c^4/(sqrt(-a^2*x^2 + 1)*a) - 7*(3*a^2*log(2*sqrt(-a^2*x^2 + 1) 
/abs(x) + 2/abs(x)) - 3*a^2/sqrt(-a^2*x^2 + 1) + 1/(sqrt(-a^2*x^2 + 1)*x^2 
))*c^4/a^3 - 2*(8*a^4*x/sqrt(-a^2*x^2 + 1) - 4*a^2/(sqrt(-a^2*x^2 + 1)*x) 
- 1/(sqrt(-a^2*x^2 + 1)*x^3))*c^4/a^4 + 11/8*(15*a^4*log(2*sqrt(-a^2*x^2 + 
 1)/abs(x) + 2/abs(x)) - 15*a^4/sqrt(-a^2*x^2 + 1) + 5*a^2/(sqrt(-a^2*x^2 
+ 1)*x^2) + 2/(sqrt(-a^2*x^2 + 1)*x^4))*c^4/a^5 - 1/5*(16*a^6*x/sqrt(-a^2* 
x^2 + 1) - 8*a^4/(sqrt(-a^2*x^2 + 1)*x) - 2*a^2/(sqrt(-a^2*x^2 + 1)*x^3) - 
 1/(sqrt(-a^2*x^2 + 1)*x^5))*c^4/a^6 - 1/16*(105*a^6*log(2*sqrt(-a^2*x^2 + 
 1)/abs(x) + 2/abs(x)) - 105*a^6/sqrt(-a^2*x^2 + 1) + 35*a^4/(sqrt(-a^2*x^ 
2 + 1)*x^2) + 14*a^2/(sqrt(-a^2*x^2 + 1)*x^4) + 8/(sqrt(-a^2*x^2 + 1)*x^6) 
)*c^4/a^7 + 1/35*(128*a^8*x/sqrt(-a^2*x^2 + 1) - 64*a^6/(sqrt(-a^2*x^2 + 1 
)*x) - 16*a^4/(sqrt(-a^2*x^2 + 1)*x^3) - 8*a^2/(sqrt(-a^2*x^2 + 1)*x^5) - 
5/(sqrt(-a^2*x^2 + 1)*x^7))*c^4/a^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (214) = 428\).

Time = 0.19 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.05 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {{\left (5 \, c^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} + \frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{4480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} - \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {15 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {\frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{4480 \, a^{6} {\left | a \right |}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="giac" 
)
 

Output:

1/4480*(5*c^4 + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) + 49*(sqrt( 
-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4*x^2) - 245*(sqrt(-a^2*x^2 + 1)*abs(a) 
 + a)^3*c^4/(a^6*x^3) - 875*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^8*x^4 
) + 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) + 9065*(sqrt(-a^2 
*x^2 + 1)*abs(a) + a)^6*c^4/(a^12*x^6))*a^14*x^7/((sqrt(-a^2*x^2 + 1)*abs( 
a) + a)^7*abs(a)) - 3*c^4*arcsin(a*x)*sgn(a)/abs(a) - 15/16*c^4*log(1/2*ab 
s(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 
 + 1)*c^4/a - 1/4480*(9065*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x + 455 
*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 875*(sqrt(-a^2*x^2 + 1)*a 
bs(a) + a)^3*c^4/x^3 - 245*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^2*x^4) 
 + 49*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^4*x^5) + 35*(sqrt(-a^2*x^2 
+ 1)*abs(a) + a)^6*c^4/(a^6*x^6) + 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7*c^4 
/(a^8*x^7))/(a^6*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.93 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {156\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^2\,x}-\frac {15\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}+\frac {62\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^4\,x^3}+\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{8\,a^5\,x^4}-\frac {6\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{16\,a} \] Input:

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

Output:

(c^4*atan((1 - a^2*x^2)^(1/2)*1i)*15i)/(16*a) - (3*c^4*asinh(x*(-a^2)^(1/2 
)))/(-a^2)^(1/2) + (c^4*(1 - a^2*x^2)^(1/2))/a - (156*c^4*(1 - a^2*x^2)^(1 
/2))/(35*a^2*x) - (15*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) + (62*c^4*(1 - 
 a^2*x^2)^(1/2))/(35*a^4*x^3) + (11*c^4*(1 - a^2*x^2)^(1/2))/(8*a^5*x^4) - 
 (6*c^4*(1 - a^2*x^2)^(1/2))/(35*a^6*x^5) - (c^4*(1 - a^2*x^2)^(1/2))/(2*a 
^7*x^6) - (c^4*(1 - a^2*x^2)^(1/2))/(7*a^8*x^7)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.77 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (-13440 \mathit {asin} \left (a x \right ) a^{7} x^{7}+4480 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-19968 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-4200 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+7936 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+6160 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-768 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2240 \sqrt {-a^{2} x^{2}+1}\, a x -640 \sqrt {-a^{2} x^{2}+1}+4200 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{7} x^{7}-4025 a^{7} x^{7}\right )}{4480 a^{8} x^{7}} \] Input:

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

Output:

(c**4*( - 13440*asin(a*x)*a**7*x**7 + 4480*sqrt( - a**2*x**2 + 1)*a**7*x** 
7 - 19968*sqrt( - a**2*x**2 + 1)*a**6*x**6 - 4200*sqrt( - a**2*x**2 + 1)*a 
**5*x**5 + 7936*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 6160*sqrt( - a**2*x**2 
+ 1)*a**3*x**3 - 768*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2240*sqrt( - a**2* 
x**2 + 1)*a*x - 640*sqrt( - a**2*x**2 + 1) + 4200*log(tan(asin(a*x)/2))*a* 
*7*x**7 - 4025*a**7*x**7))/(4480*a**8*x**7)