\(\int e^{-3 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\) [704]

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 = 0.16 (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[(c - c/(a^2*x^2))^4/E^(3*ArcTanh[a*x]),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 = 0.66 (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, 6699, 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

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

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(c - c/(a^2*x^2))^4/E^(3*ArcTanh[a*x]),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 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 = 0.55 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {\left (560 a^{9} x^{9}+2496 a^{8} x^{8}-1085 a^{7} x^{7}-3488 x^{6} a^{6}+1295 a^{5} x^{5}+1088 a^{4} x^{4}-1050 a^{3} x^{3}-16 a^{2} x^{2}+280 a x -80\right ) c^{4}}{560 x^{7} \sqrt {-a^{2} x^{2}+1}\, a^{8}}-\frac {\left (-\frac {15 a^{7} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {3 a^{8} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{4}}{a^{8}}\)	\(152\)
default	\(\frac {c^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{7 x^{7}}-\frac {16 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{35 x^{5}}+a^{5} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{6 x^{6}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )}{6}\right )+2 a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )-3 a^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )}{3}\right )\right )}{a^{8}}\)	\(431\)

Input:

int((c-c/a^2/x^2)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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+108 
8*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 = 0.11 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.72 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 525 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 560 \, a^{7} c^{4} x^{7} - {\left (560 \, a^{7} c^{4} x^{7} + 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} - 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} + 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x + 80 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}} \] Input:

integrate((c-c/a^2/x^2)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.70 (sec) , antiderivative size = 1110, normalized size of antiderivative = 4.51 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\text {Too large to display} \] Input:

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

Output:

-c**4*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*a 
sin(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 + 3*c**4*Piecewise((-I*a**2*x/ 
sqrt(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 - c**4*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*s 
qrt(-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 - 5*c**4*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqr 
t(-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 + 5*c** 
4*Piecewise((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* 
sqrt(1 - 1/(a**2*x**2))), True))/a**5 + c**4*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), Tru 
e))/a**6 - 3*c**4*Piecewise((a**6*acosh(1/(a*x))/16 - a**5/(16*x*sqrt(-...
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (214) = 428\).

Time = 0.14 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.06 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {{\left (5 \, c^{4} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} + \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} - \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} + \frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{4480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} - \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {15 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {\frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} - \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} + \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{4480 \, a^{6} {\left | a \right |}} \] Input:

integrate((c-c/a^2/x^2)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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 = 14.71 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.93 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {15\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {156\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^2\,x}-\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {62\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^4\,x^3}-\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{8\,a^5\,x^4}-\frac {6\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}-\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{16\,a} \] Input:

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

Output:

(15*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) - (c^4*atan((1 - a^2*x^2)^(1/2)* 
1i)*15i)/(16*a) - (c^4*(1 - a^2*x^2)^(1/2))/a - (156*c^4*(1 - a^2*x^2)^(1/ 
2))/(35*a^2*x) - (3*c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (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((c-c/a^2/x^2)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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)