\(\int \frac {e^{2 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [727]

Optimal result

Integrand size = 24, antiderivative size = 238 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {2}{5 a c \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2}{3 a c^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2}{a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \left (a+\frac {1}{x}\right )}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {19}{35 a^2 c \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {37}{35 a^2 c^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {109}{35 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{c^4}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a c^{7/2}} \] Output:

2/5/a/c/(c-c/a^2/x^2)^(5/2)+2/3/a/c^2/(c-c/a^2/x^2)^(3/2)+2/a/c^3/(c-c/a^2 
/x^2)^(1/2)+2/7*(a+1/x)/a^2/(c-c/a^2/x^2)^(7/2)+19/35/a^2/c/(c-c/a^2/x^2)^ 
(5/2)/x+37/35/a^2/c^2/(c-c/a^2/x^2)^(3/2)/x+109/35/a^2/c^3/(c-c/a^2/x^2)^( 
1/2)/x-(c-c/a^2/x^2)^(1/2)*x/c^4-2*arctanh((c-c/a^2/x^2)^(1/2)/c^(1/2))/a/ 
c^(7/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.56 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {-432+654 a x+636 a^2 x^2-1226 a^3 x^3-74 a^4 x^4+562 a^5 x^5-105 a^6 x^6-210 (-1+a x)^3 (1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{105 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^3 (1+a x)} \] Input:

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

Output:

(-432 + 654*a*x + 636*a^2*x^2 - 1226*a^3*x^3 - 74*a^4*x^4 + 562*a^5*x^5 - 
105*a^6*x^6 - 210*(-1 + a*x)^3*(1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt 
[-1 + a^2*x^2]])/(105*a^2*c^3*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)^3*(1 + a* 
x))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6709, 529, 2166, 2345, 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.

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

\(\Big \downarrow \) 6709

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\frac {35 x^5}{a^2}+\frac {70 x^4}{a^3}+\frac {105 x^3}{a^4}+\frac {140 x^2}{a^5}+\frac {175 x}{a^6}+\frac {34}{a^7}}{\left (1-a^2 x^2\right )^{5/2}}dx-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {244 a x+315}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {\frac {105 x^3}{a^4}+\frac {210 x^2}{a^5}+\frac {420 x}{a^6}+\frac {142}{a^7}}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {105 (a x+2)}{a^7 \sqrt {1-a^2 x^2}}dx-\frac {352 a x+525}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {244 a x+315}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 \int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a^7}-\frac {352 a x+525}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {244 a x+315}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^7}-\frac {352 a x+525}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {244 a x+315}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {(a x+1)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}+\frac {1}{7} \left (\frac {1}{5} \left (\frac {244 a x+315}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{3} \left (\frac {105 \left (\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^7}-\frac {352 a x+525}{a^8 \sqrt {1-a^2 x^2}}\right )\right )-\frac {44 (a x+1)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

Input:

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

Output:

((1 - a^2*x^2)^(7/2)*((1 + a*x)^2/(7*a^8*(1 - a^2*x^2)^(7/2)) + ((-44*(1 + 
 a*x))/(5*a^8*(1 - a^2*x^2)^(5/2)) + ((315 + 244*a*x)/(3*a^8*(1 - a^2*x^2) 
^(3/2)) + (-((525 + 352*a*x)/(a^8*Sqrt[1 - a^2*x^2])) + (105*(-(Sqrt[1 - a 
^2*x^2]/a) + (2*ArcSin[a*x])/a))/a^7)/3)/5)/7))/((c - c/(a^2*x^2))^(7/2)*x 
^7)
 

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 = 0.24 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.61

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a^{2} c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}-\frac {\left (\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{7} \sqrt {a^{2} c}}-\frac {3061 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{840 a^{9} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{28 a^{12} c \left (x -\frac {1}{a}\right )^{4}}-\frac {39 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{140 a^{11} c \left (x -\frac {1}{a}\right )^{3}}-\frac {1753 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{1680 a^{10} c \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{48 a^{10} c \left (x +\frac {1}{a}\right )^{2}}+\frac {7 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{24 a^{9} c \left (x +\frac {1}{a}\right )}\right ) a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\)	\(384\)
default	\(-\frac {\left (105 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{7} x^{7}+96 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{6} x^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}-96 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{5} x^{5}-392 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{5} x^{5}-240 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{4} x^{4}+1540 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{4} x^{4}+210 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{6} c x +240 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{3} x^{3}+350 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{3} x^{3}-210 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{5} c +180 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{2} x^{2}-1470 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{2} x^{2}-180 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a x -42 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a x -30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}}+462 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x +1\right )}{105 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} a^{8} c^{\frac {7}{2}}}\)	\(572\)

Input:

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

Output:

-1/a^2*(a^2*x^2-1)/c^3/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)-(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)-3061/840/a^9/c/(x-1/a)*((x 
-1/a)^2*a^2*c+2*(x-1/a)*a*c)^(1/2)-1/28/a^12/c/(x-1/a)^4*((x-1/a)^2*a^2*c+ 
2*(x-1/a)*a*c)^(1/2)-39/140/a^11/c/(x-1/a)^3*((x-1/a)^2*a^2*c+2*(x-1/a)*a* 
c)^(1/2)-1753/1680/a^10/c/(x-1/a)^2*((x-1/a)^2*a^2*c+2*(x-1/a)*a*c)^(1/2)- 
1/48/a^10/c/(x+1/a)^2*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2)+7/24/a^9/c/(x+ 
1/a)*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2))*a^6/c^3/x/(c*(a^2*x^2-1)/a^2/x 
^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.09 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\left [\frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {210 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:

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

Output:

[1/105*(105*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 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) - (105*a^7*x^7 - 562*a^6*x^6 + 74*a^5*x^5 + 1226*a^4*x^4 - 636* 
a^3*x^3 - 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4 
*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c 
^4*x + a*c^4), 1/105*(210*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2 
*x^2 - 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)) - (105*a^7*x^7 - 562*a^6*x^6 + 74*a^5*x^5 + 1226 
*a^4*x^4 - 636*a^3*x^3 - 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2* 
x^2)))/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^ 
4*x^2 - 2*a^2*c^4*x + a*c^4)]
 

Sympy [F]

\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=- \int \frac {a x}{a c^{3} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx - \int \frac {1}{a c^{3} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {3 c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {c^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx \] Input:

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

Output:

-Integral(a*x/(a*c**3*x*sqrt(c - c/(a**2*x**2)) - c**3*sqrt(c - c/(a**2*x* 
*2)) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a*x) + 3*c**3*sqrt(c - c/(a**2*x**2 
))/(a**2*x**2) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 3*c**3*sqrt( 
c - c/(a**2*x**2))/(a**4*x**4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) 
+ c**3*sqrt(c - c/(a**2*x**2))/(a**6*x**6)), x) - Integral(1/(a*c**3*x*sqr 
t(c - c/(a**2*x**2)) - c**3*sqrt(c - c/(a**2*x**2)) - 3*c**3*sqrt(c - c/(a 
**2*x**2))/(a*x) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 3*c**3*sqr 
t(c - c/(a**2*x**2))/(a**3*x**3) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a**4*x* 
*4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + c**3*sqrt(c - c/(a**2*x** 
2))/(a**6*x**6)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.55 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-420 \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}+2248 \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}-296 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-4904 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+2544 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+2616 \sqrt {a^{2} x^{2}-1}\, a x -1728 \sqrt {a^{2} x^{2}-1}-840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{6} x^{6}+1680 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{5} x^{5}+840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{4} x^{4}-3360 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{3} x^{3}+840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{2} x^{2}+1680 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x -840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )-463 a^{6} x^{6}+926 a^{5} x^{5}+463 a^{4} x^{4}-1852 a^{3} x^{3}+463 a^{2} x^{2}+926 a x -463\right )}{420 a \,c^{4} \left (a^{6} x^{6}-2 a^{5} x^{5}-a^{4} x^{4}+4 a^{3} x^{3}-a^{2} x^{2}-2 a x +1\right )} \] Input:

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

Output:

(sqrt(c)*( - 420*sqrt(a**2*x**2 - 1)*a**6*x**6 + 2248*sqrt(a**2*x**2 - 1)* 
a**5*x**5 - 296*sqrt(a**2*x**2 - 1)*a**4*x**4 - 4904*sqrt(a**2*x**2 - 1)*a 
**3*x**3 + 2544*sqrt(a**2*x**2 - 1)*a**2*x**2 + 2616*sqrt(a**2*x**2 - 1)*a 
*x - 1728*sqrt(a**2*x**2 - 1) - 840*log(sqrt(a**2*x**2 - 1) + a*x)*a**6*x* 
*6 + 1680*log(sqrt(a**2*x**2 - 1) + a*x)*a**5*x**5 + 840*log(sqrt(a**2*x** 
2 - 1) + a*x)*a**4*x**4 - 3360*log(sqrt(a**2*x**2 - 1) + a*x)*a**3*x**3 + 
840*log(sqrt(a**2*x**2 - 1) + a*x)*a**2*x**2 + 1680*log(sqrt(a**2*x**2 - 1 
) + a*x)*a*x - 840*log(sqrt(a**2*x**2 - 1) + a*x) - 463*a**6*x**6 + 926*a* 
*5*x**5 + 463*a**4*x**4 - 1852*a**3*x**3 + 463*a**2*x**2 + 926*a*x - 463)) 
/(420*a*c**4*(a**6*x**6 - 2*a**5*x**5 - a**4*x**4 + 4*a**3*x**3 - a**2*x** 
2 - 2*a*x + 1))