\(\int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a^2 c x^2)^3} \, dx\) [939]

Optimal result

Integrand size = 23, antiderivative size = 200 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {7 a^2}{6 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3 a^3 x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {7 a^2}{2 c^3 \sqrt {1-a^2 x^2}}+\frac {11 a^3 x}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {7 a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \] Output:

1/5*a^2*(a*x+1)/c^3/(-a^2*x^2+1)^(5/2)+7/6*a^2/c^3/(-a^2*x^2+1)^(3/2)-1/2/ 
c^3/x^2/(-a^2*x^2+1)^(3/2)+3/5*a^3*x/c^3/(-a^2*x^2+1)^(3/2)+7/2*a^2/c^3/(- 
a^2*x^2+1)^(1/2)+11/5*a^3*x/c^3/(-a^2*x^2+1)^(1/2)-a*(-a^2*x^2+1)^(1/2)/c^ 
3/x-7/2*a^2*arctanh((-a^2*x^2+1)^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {-15-15 a x+176 a^2 x^2+4 a^3 x^3-249 a^4 x^4+9 a^5 x^5+96 a^6 x^6-105 a^2 x^2 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{30 c^3 x^2 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(-15 - 15*a*x + 176*a^2*x^2 + 4*a^3*x^3 - 249*a^4*x^4 + 9*a^5*x^5 + 96*a^6 
*x^6 - 105*a^2*x^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 
 - a^2*x^2]])/(30*c^3*x^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.77, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6698, 532, 25, 2336, 27, 2336, 27, 2338, 25, 27, 534, 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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 532

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {1}{5} \left (\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {3 \left (6 a^3 x^3+10 a^2 x^2+5 a x+5\right )}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\int \frac {6 a^3 x^3+10 a^2 x^2+5 a x+5}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {1}{5} \left (-\int -\frac {5 \left (3 a^2 x^2+a x+1\right )}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (5 \int \frac {3 a^2 x^2+a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (-\frac {1}{2} \int -\frac {a (7 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} \int \frac {a (7 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \int \frac {7 a x+2}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \left (7 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \left (\frac {7}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \left (-\frac {7 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \left (-7 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (11 a x+15)}{\sqrt {1-a^2 x^2}}+\frac {a^2 (9 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

Input:

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

Output:

((a^2*(1 + a*x))/(5*(1 - a^2*x^2)^(5/2)) + ((a^2*(10 + 9*a*x))/(3*(1 - a^2 
*x^2)^(3/2)) + (a^2*(15 + 11*a*x))/Sqrt[1 - a^2*x^2] + 5*(-1/2*Sqrt[1 - a^ 
2*x^2]/x^2 + (a*((-2*Sqrt[1 - a^2*x^2])/x - 7*a*ArcTanh[Sqrt[1 - a^2*x^2]] 
))/2))/5)/c^3
 

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 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 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*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))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

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]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {7 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 a \left (x -\frac {1}{a}\right )^{3}}-\frac {11 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{10}+\frac {39 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 \left (x -\frac {1}{a}\right )}+\frac {a \left (-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}\right )}{8}-\frac {9 a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 \left (x +\frac {1}{a}\right )}}{c^{3}}\)	\(326\)
risch	\(\frac {2 a^{3} x^{3}+a^{2} x^{2}-2 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {a^{2} \left (-7 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{2 a^{2}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {39 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{8 a \left (x -\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{4 a}+\frac {9 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{8 a \left (x +\frac {1}{a}\right )}\right )}{2 c^{3}}\)	\(431\)

Input:

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

Output:

-1/c^3*(1/2*(-a^2*x^2+1)^(1/2)/x^2+7/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))+a 
*(-a^2*x^2+1)^(1/2)/x+1/20/a/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)- 
11/10*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(- 
(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))+39/16*a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1 
/a))^(1/2)+1/8*a*(-1/3/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/ 
(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))-9/16*a/(x+1/a)*(-a^2*(x+1/a)^2 
+2*a*(x+1/a))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {116 \, a^{7} x^{7} - 116 \, a^{6} x^{6} - 232 \, a^{5} x^{5} + 232 \, a^{4} x^{4} + 116 \, a^{3} x^{3} - 116 \, a^{2} x^{2} + 105 \, {\left (a^{7} x^{7} - a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{4} x^{4} + a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (96 \, a^{6} x^{6} + 9 \, a^{5} x^{5} - 249 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 176 \, a^{2} x^{2} - 15 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{5} c^{3} x^{7} - a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{4} + a c^{3} x^{3} - c^{3} x^{2}\right )}} \] Input:

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

Output:

1/30*(116*a^7*x^7 - 116*a^6*x^6 - 232*a^5*x^5 + 232*a^4*x^4 + 116*a^3*x^3 
- 116*a^2*x^2 + 105*(a^7*x^7 - a^6*x^6 - 2*a^5*x^5 + 2*a^4*x^4 + a^3*x^3 - 
 a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (96*a^6*x^6 + 9*a^5*x^5 - 249* 
a^4*x^4 + 4*a^3*x^3 + 176*a^2*x^2 - 15*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5* 
c^3*x^7 - a^4*c^3*x^6 - 2*a^3*c^3*x^5 + 2*a^2*c^3*x^4 + a*c^3*x^3 - c^3*x^ 
2)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {a}{- a^{6} x^{8} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{9} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

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

Output:

(Integral(a/(-a**6*x**8*sqrt(-a**2*x**2 + 1) + 3*a**4*x**6*sqrt(-a**2*x**2 
 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) 
+ Integral(1/(-a**6*x**9*sqrt(-a**2*x**2 + 1) + 3*a**4*x**7*sqrt(-a**2*x** 
2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) 
)/c**3
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.78 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {11\,a^4\,\sqrt {1-a^2\,x^2}}{30\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^4\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,x^2}-\frac {a\,\sqrt {1-a^2\,x^2}}{c^3\,x}-\frac {29\,a^3\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {673\,a^3\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{2\,c^3} \] Input:

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

Output:

(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*7i)/(2*c^3) + (11*a^4*(1 - a^2*x^2)^(1/2 
))/(30*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (a^4*(1 - a^2*x^2)^(1/2))/ 
(24*(a^2*c^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(2*c^3*x^ 
2) - (a*(1 - a^2*x^2)^(1/2))/(c^3*x) - (29*a^3*(1 - a^2*x^2)^(1/2))/(48*(- 
a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a)) + (673*a^3*(1 - a^ 
2*x^2)^(1/2))/(240*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a 
)) + (a^3*(1 - a^2*x^2)^(1/2))/(20*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c 
^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx=\frac {-192 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-18 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+498 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-8 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-352 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+30 \sqrt {-a^{2} x^{2}+1}\, a x +30 \sqrt {-a^{2} x^{2}+1}+105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{7} x^{7}-105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{6} x^{6}-210 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{5} x^{5}+210 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{4} x^{4}+105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}-105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{2} x^{2}-105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{7} x^{7}+105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{6} x^{6}+210 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{5} x^{5}-210 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{4} x^{4}-105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}+105 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{2} x^{2}}{60 c^{3} x^{2} \left (a^{5} x^{5}-a^{4} x^{4}-2 a^{3} x^{3}+2 a^{2} x^{2}+a x -1\right )} \] Input:

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

Output:

( - 192*sqrt( - a**2*x**2 + 1)*a**6*x**6 - 18*sqrt( - a**2*x**2 + 1)*a**5* 
x**5 + 498*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 8*sqrt( - a**2*x**2 + 1)*a** 
3*x**3 - 352*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 30*sqrt( - a**2*x**2 + 1)* 
a*x + 30*sqrt( - a**2*x**2 + 1) + 105*log(sqrt( - a**2*x**2 + 1) - 1)*a**7 
*x**7 - 105*log(sqrt( - a**2*x**2 + 1) - 1)*a**6*x**6 - 210*log(sqrt( - a* 
*2*x**2 + 1) - 1)*a**5*x**5 + 210*log(sqrt( - a**2*x**2 + 1) - 1)*a**4*x** 
4 + 105*log(sqrt( - a**2*x**2 + 1) - 1)*a**3*x**3 - 105*log(sqrt( - a**2*x 
**2 + 1) - 1)*a**2*x**2 - 105*log(sqrt( - a**2*x**2 + 1) + 1)*a**7*x**7 + 
105*log(sqrt( - a**2*x**2 + 1) + 1)*a**6*x**6 + 210*log(sqrt( - a**2*x**2 
+ 1) + 1)*a**5*x**5 - 210*log(sqrt( - a**2*x**2 + 1) + 1)*a**4*x**4 - 105* 
log(sqrt( - a**2*x**2 + 1) + 1)*a**3*x**3 + 105*log(sqrt( - a**2*x**2 + 1) 
 + 1)*a**2*x**2)/(60*c**3*x**2*(a**5*x**5 - a**4*x**4 - 2*a**3*x**3 + 2*a* 
*2*x**2 + a*x - 1))