\(\int \frac {e^{2 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{5/2}} \, dx\) [726]

Optimal result

Integrand size = 24, antiderivative size = 185 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {2}{3 a c \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {13}{15 a^2 c \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {41}{15 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{c^3}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a c^{5/2}} \] Output:

2/3/a/c/(c-c/a^2/x^2)^(3/2)+2/a/c^2/(c-c/a^2/x^2)^(1/2)+2/5*(a+1/x)/a^2/(c 
-c/a^2/x^2)^(5/2)+13/15/a^2/c/(c-c/a^2/x^2)^(3/2)/x+41/15/a^2/c^2/(c-c/a^2 
/x^2)^(1/2)/x-(c-c/a^2/x^2)^(1/2)*x/c^3-2*arctanh((c-c/a^2/x^2)^(1/2)/c^(1 
/2))/a/c^(5/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.57 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {56-82 a x-32 a^2 x^2+76 a^3 x^3-15 a^4 x^4-30 (-1+a x)^2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{15 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^2} \] Input:

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2),x]
 

Output:

(56 - 82*a*x - 32*a^2*x^2 + 76*a^3*x^3 - 15*a^4*x^4 - 30*(-1 + a*x)^2*Sqrt 
[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(15*a^2*c^2*Sqrt[c - c/(a^2* 
x^2)]*x*(-1 + a*x)^2)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6709, 529, 2166, 2345, 27, 455, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6709

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\frac {15 x^3}{a^2}+\frac {30 x^2}{a^3}+\frac {45 x}{a^4}+\frac {16}{a^5}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\int \frac {15 (a x+2)}{a^5 \sqrt {1-a^2 x^2}}dx\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \left (\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\)

Input:

Int[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2),x]
 

Output:

((1 - a^2*x^2)^(5/2)*((1 + a*x)^2/(5*a^6*(1 - a^2*x^2)^(5/2)) + ((-22*(1 + 
 a*x))/(3*a^6*(1 - a^2*x^2)^(3/2)) + ((2*(30 + 23*a*x))/(a^6*Sqrt[1 - a^2* 
x^2]) - (15*(-(Sqrt[1 - a^2*x^2]/a) + (2*ArcSin[a*x])/a))/a^5)/3)/5))/((c 
- c/(a^2*x^2))^(5/2)*x^5)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 455

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

rule 529

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

rule 2166

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

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[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 = 0.23 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a^{2} c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}-\frac {\left (\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{5} \sqrt {a^{2} c}}-\frac {383 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{120 a^{7} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{10 a^{9} c \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{60 a^{8} c \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{8 a^{7} c \left (x +\frac {1}{a}\right )}\right ) a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\)	\(300\)
default	\(-\frac {\left (15 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{5} x^{5}-16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{4} x^{4}-45 x^{4} c^{\frac {5}{2}} a^{4} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}+16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{3} x^{3}-60 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{3} x^{3}+30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{4} c x +24 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{2} x^{2}+90 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{2} x^{2}-30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{3} c -24 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a x +50 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a x -6 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}}-50 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}\right ) \left (a x +1\right )}{15 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} a^{6} c^{\frac {5}{2}}}\)	\(462\)

Input:

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/a^2*(a^2*x^2-1)/c^2/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x-(2/a^5*ln(a^2*c*x/( 
a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)-383/120/a^7/c/(x-1/a)*((x- 
1/a)^2*a^2*c+2*(x-1/a)*a*c)^(1/2)-1/10/a^9/c/(x-1/a)^3*((x-1/a)^2*a^2*c+2* 
(x-1/a)*a*c)^(1/2)-41/60/a^8/c/(x-1/a)^2*((x-1/a)^2*a^2*c+2*(x-1/a)*a*c)^( 
1/2)+1/8/a^7/c/(x+1/a)*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2))*a^4/c^2*(c*( 
a^2*x^2-1))^(1/2)/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.91 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {30 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

[1/15*(15*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(c)*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) - (15*a^5*x^5 - 76*a^4* 
x^4 + 32*a^3*x^3 + 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/( 
a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3), 1/15*(30*(a^4*x^4 - 2* 
a^3*x^3 + 2*a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c) 
/(a^2*x^2))/(a^2*c*x^2 - c)) - (15*a^5*x^5 - 76*a^4*x^4 + 32*a^3*x^3 + 82* 
a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 - 2*a^4*c^ 
3*x^3 + 2*a^2*c^3*x - a*c^3)]
 

Sympy [F]

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

integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(5/2),x)
 

Output:

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

Maxima [F]

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

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

-integrate((a*x + 1)^2/((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(5/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a^2\,x^2-1\right )} \,d x \] Input:

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(5/2)*(a^2*x^2 - 1)),x)
 

Output:

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(5/2)*(a^2*x^2 - 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-30 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+152 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-64 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-164 \sqrt {a^{2} x^{2}-1}\, a x +112 \sqrt {a^{2} x^{2}-1}-60 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{4} x^{4}+120 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{3} x^{3}-120 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x +60 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )-47 a^{4} x^{4}+94 a^{3} x^{3}-94 a x +47\right )}{30 a \,c^{3} \left (a^{4} x^{4}-2 a^{3} x^{3}+2 a x -1\right )} \] Input:

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x)
 

Output:

(sqrt(c)*( - 30*sqrt(a**2*x**2 - 1)*a**4*x**4 + 152*sqrt(a**2*x**2 - 1)*a* 
*3*x**3 - 64*sqrt(a**2*x**2 - 1)*a**2*x**2 - 164*sqrt(a**2*x**2 - 1)*a*x + 
 112*sqrt(a**2*x**2 - 1) - 60*log(sqrt(a**2*x**2 - 1) + a*x)*a**4*x**4 + 1 
20*log(sqrt(a**2*x**2 - 1) + a*x)*a**3*x**3 - 120*log(sqrt(a**2*x**2 - 1) 
+ a*x)*a*x + 60*log(sqrt(a**2*x**2 - 1) + a*x) - 47*a**4*x**4 + 94*a**3*x* 
*3 - 94*a*x + 47))/(30*a*c**3*(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1))