\(\int e^{-2 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\) [759]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {c \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (12 a+\frac {25}{x}\right )}{30 a^2}-\frac {c^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (16 a+\frac {25}{x}\right )}{24 a^2}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (32 a+\frac {25}{x}\right )}{16 a^2}-\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x-\frac {25 c^{7/2} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right )}{16 a}+\frac {2 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a} \] Output:

-1/30*c*(c-c/a^2/x^2)^(5/2)*(12*a+25/x)/a^2-1/24*c^2*(c-c/a^2/x^2)^(3/2)*( 
16*a+25/x)/a^2-1/16*c^3*(c-c/a^2/x^2)^(1/2)*(32*a+25/x)/a^2-(c-c/a^2/x^2)^ 
(7/2)*x-25/16*c^(7/2)*arctan(c^(1/2)/a/(c-c/a^2/x^2)^(1/2)/x)/a+2*c^(7/2)* 
arctanh((c-c/a^2/x^2)^(1/2)/c^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-40+96 a x+70 a^2 x^2-352 a^3 x^3+105 a^4 x^4+736 a^5 x^5+240 a^6 x^6\right )+375 a^6 x^6 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )-480 a^6 x^6 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{240 a^6 x^5 \sqrt {-1+a^2 x^2}} \] Input:

Integrate[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

-1/240*(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 + 96*a*x + 70*a 
^2*x^2 - 352*a^3*x^3 + 105*a^4*x^4 + 736*a^5*x^5 + 240*a^6*x^6) + 375*a^6* 
x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x^2 
]]))/(a^6*x^5*Sqrt[-1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6709, 570, 540, 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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx\)

\(\Big \downarrow \) 6709

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \int \frac {\left (1-a^2 x^2\right )^{9/2}}{x^7 (a x+1)^2}dx}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 540

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} \int \frac {a (12-5 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \int \frac {(12-5 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (\frac {1}{4} a^2 \int -\frac {(48-25 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \int \frac {(48-25 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (32-25 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(32-25 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 536

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {-32 x a^2-25 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(25 a x+32) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-32 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-25 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(25 a x+32) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-25 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (25 a x+32)}{x}-32 a \arcsin (a x)\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {25}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (25 a x+32)}{x}-32 a \arcsin (a x)\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {25 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (25 a x+32)}{x}-32 a \arcsin (a x)\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (25 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (25 a x+32)}{x}-32 a \arcsin (a x)\right )-\frac {(32-25 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(48-25 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\)

Input:

Int[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

((c - c/(a^2*x^2))^(7/2)*x^7*(-1/6*(1 - a^2*x^2)^(7/2)/x^6 - (a*(-1/20*((4 
8 - 25*a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/2*((32 - 25*a*x)*(1 - a^2* 
x^2)^(3/2))/x^3 - (3*a^2*(-(((32 + 25*a*x)*Sqrt[1 - a^2*x^2])/x) - 32*a*Ar 
cSin[a*x] + 25*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/6))/(1 - a^2*x^2)^(7 
/2)
 

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 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 6709

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.31

method	result	size
risch	\(-\frac {\left (736 a^{7} x^{7}+105 x^{6} a^{6}-1088 a^{5} x^{5}-35 a^{4} x^{4}+448 a^{3} x^{3}-110 a^{2} x^{2}-96 a x +40\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{240 x^{5} a^{6} \left (a^{2} x^{2}-1\right )}-\frac {\left (\frac {25 a^{6} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{16 \sqrt {-c}}-\frac {2 a^{7} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{6} \left (a^{2} x^{2}-1\right )}\)	\(251\)
default	\(-\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} x \left (-2016 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{9} c \,x^{7}+2016 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{9} x^{5}-375 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{8} c \,x^{6}+480 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{8} c \,x^{6}-105 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{8} x^{4}+2352 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{7} c^{2} x^{7}-560 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{7} c^{2} x^{7}+224 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{7} x^{3}+525 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{6} c^{2} x^{6}-2940 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} c^{3} x^{7}+700 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} c^{3} x^{7}-630 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{6} x^{2}-875 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{4} c^{3} x^{6}+672 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} x +4410 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c^{4} x^{7}-1050 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c^{4} x^{7}-280 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}+2625 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c^{4} x^{6}-4410 c^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{6}+1050 c^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) a \,x^{6}+2625 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{5} x^{6}\right )}{1680 a^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, c}\)	\(795\)

Input:

int((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

-1/240*(736*a^7*x^7+105*a^6*x^6-1088*a^5*x^5-35*a^4*x^4+448*a^3*x^3-110*a^ 
2*x^2-96*a*x+40)/x^5*c^3/a^6*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)-(25 
/16*a^6/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*(a^2*c*x^2-c)^(1/2))/x)-2*a^7*ln( 
a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+a^6/c*(c*(a^2*x^2 
-1))^(1/2))*c^3/a^6*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)*x*(c*(a^2*x^ 
2-1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.23 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\left [-\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, \frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) + 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \] Input:

integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas" 
)
 

Output:

[-1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 
- c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 375*a^5*sqrt(-c)*c^3*x^5*log(-(a^2*c*x^ 
2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(240*a^ 
6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c 
^3*x^2 + 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), 
1/240*(375*a^5*c^(7/2)*x^5*arctan(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/sqrt 
(c)) + 240*a^5*c^(7/2)*x^5*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c 
*x^2 - c)/(a^2*x^2)) - c) - (240*a^6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c 
^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 + 96*a*c^3*x - 40*c^3)*sqrt((a^2 
*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 18.08 (sec) , antiderivative size = 1059, normalized size of antiderivative = 5.54 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\text {Too large to display} \] Input:

integrate((c-c/a**2/x**2)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

-c**3*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I* 
sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1 
), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I* 
sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**3*Piecewise((-a*sqr 
t(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x 
**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sq 
rt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + c**3*Pi 
ecewise((I*a*sqrt(c)*acosh(1/(a*x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x 
**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2 
) > 1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x 
), True))/a**2 - 4*c**3*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2)) 
**(3/2)/(3*c), True))/a**3 + c**3*Piecewise((I*a**3*sqrt(c)*acosh(1/(a*x)) 
/8 - I*a**2*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*s 
qrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(4*a**2*x**5*sqrt(-1 + 1/(a**2*x**2)) 
), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*sqrt(c)/(8 
*x*sqrt(1 - 1/(a**2*x**2))) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + 
 sqrt(c)/(4*a**2*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**4 + 2*c**3*Piece 
wise((2*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + a*sqrt(c)*sqrt(a**2*x**2 
 - 1)/(15*x**3) - sqrt(c)*sqrt(a**2*x**2 - 1)/(5*a*x**5), Abs(a**2*x**2) > 
 1), (2*I*a**3*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-...
 

Maxima [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima" 
)
 

Output:

-integrate((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(7/2)/(a*x + 1)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (163) = 326\).

Time = 0.28 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.94 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {1}{120} \, {\left (\frac {375 \, c^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {240 \, c^{\frac {7}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {120 \, \sqrt {a^{2} c x^{2} - c} c^{3} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 1440 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 4320 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 7360 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 6720 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) - 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 2976 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 736 \, a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \] Input:

integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

1/120*(375*c^(7/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))* 
sgn(x)/a^2 - 240*c^(7/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sg 
n(x)/(a*abs(a)) - 120*sqrt(a^2*c*x^2 - c)*c^3*sgn(x)/a^2 + (105*(sqrt(a^2* 
c)*x - sqrt(a^2*c*x^2 - c))^11*c^4*abs(a)*sgn(x) - 1440*(sqrt(a^2*c)*x - s 
qrt(a^2*c*x^2 - c))^10*a*c^(9/2)*sgn(x) + 595*(sqrt(a^2*c)*x - sqrt(a^2*c* 
x^2 - c))^9*c^5*abs(a)*sgn(x) - 4320*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c)) 
^8*a*c^(11/2)*sgn(x) - 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^6*abs 
(a)*sgn(x) - 7360*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(13/2)*sgn(x 
) + 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^7*abs(a)*sgn(x) - 6720*( 
sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(15/2)*sgn(x) - 595*(sqrt(a^2*c 
)*x - sqrt(a^2*c*x^2 - c))^3*c^8*abs(a)*sgn(x) - 2976*(sqrt(a^2*c)*x - sqr 
t(a^2*c*x^2 - c))^2*a*c^(17/2)*sgn(x) - 105*(sqrt(a^2*c)*x - sqrt(a^2*c*x^ 
2 - c))*c^9*abs(a)*sgn(x) - 736*a*c^(19/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt 
(a^2*c*x^2 - c))^2 + c)^6*a^2*abs(a)))*abs(a)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:

int(-((c - c/(a^2*x^2))^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
 

Output:

-int(((c - c/(a^2*x^2))^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (750 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{6} x^{6}-240 \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}-736 \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}-105 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+352 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-70 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-96 \sqrt {a^{2} x^{2}-1}\, a x +40 \sqrt {a^{2} x^{2}-1}+480 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{6} x^{6}+256 a^{6} x^{6}\right )}{240 a^{7} x^{6}} \] Input:

int((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(sqrt(c)*c**3*(750*atan(sqrt(a**2*x**2 - 1) + a*x)*a**6*x**6 - 240*sqrt(a* 
*2*x**2 - 1)*a**6*x**6 - 736*sqrt(a**2*x**2 - 1)*a**5*x**5 - 105*sqrt(a**2 
*x**2 - 1)*a**4*x**4 + 352*sqrt(a**2*x**2 - 1)*a**3*x**3 - 70*sqrt(a**2*x* 
*2 - 1)*a**2*x**2 - 96*sqrt(a**2*x**2 - 1)*a*x + 40*sqrt(a**2*x**2 - 1) + 
480*log(sqrt(a**2*x**2 - 1) + a*x)*a**6*x**6 + 256*a**6*x**6))/(240*a**7*x 
**6)