\(\int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\) [941]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \] Output:

1/3*a*(a*x+1)/c^2/(-a^2*x^2+1)^(3/2)+1/3*(3*a*x+5)/c^2/x/(-a^2*x^2+1)^(1/2 
)-8/3*(-a^2*x^2+1)^(1/2)/c^2/x-a*arctanh((-a^2*x^2+1)^(1/2))/c^2
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {3-7 a x-5 a^2 x^2+8 a^3 x^3-3 a x (-1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{3 c^2 x (-1+a x) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(3 - 7*a*x - 5*a^2*x^2 + 8*a^3*x^3 - 3*a*x*(-1 + a*x)*Sqrt[1 - a^2*x^2]*Ar 
cTanh[Sqrt[1 - a^2*x^2]])/(3*c^2*x*(-1 + a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6698, 532, 25, 2336, 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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 532

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2336

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((a*(1 + a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a*(3 + 5*a*x))/Sqrt[1 - a^2*x^2 
] + 3*(-(Sqrt[1 - a^2*x^2]/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]))/3)/c^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 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 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 = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.46

method	result	size
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 a \left (x -\frac {1}{a}\right )^{2}}-\frac {17 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 \left (x +\frac {1}{a}\right )}}{c^{2}}\)	\(150\)
risch	\(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {a \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a \left (x +\frac {1}{a}\right )}\right )}{c^{2}}\)	\(211\)

Input:

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

Output:

1/c^2*(-(-a^2*x^2+1)^(1/2)/x-a*arctanh(1/(-a^2*x^2+1)^(1/2))+1/6/a/(x-1/a) 
^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-17/12/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x 
-1/a))^(1/2)-1/4/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.49 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {4 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x^{4} - a^{2} c^{2} x^{3} - a c^{2} x^{2} + c^{2} x\right )}} \] Input:

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

Output:

1/3*(4*a^4*x^4 - 4*a^3*x^3 - 4*a^2*x^2 + 4*a*x + 3*(a^4*x^4 - a^3*x^3 - a^ 
2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^3*x^3 - 5*a^2*x^2 - 7* 
a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^3*c^2*x^4 - a^2*c^2*x^3 - a*c^2*x^2 + c^2* 
x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=-\frac {\frac {3 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c^{2}} - \frac {3 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c^{2}} + \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} - 1\right )} a^{2} - a^{2}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}}{6 \, a} + \frac {8 \, a^{4} x^{4} - 12 \, a^{2} x^{2} + 3}{3 \, {\left (a^{2} c^{2} x^{3} - c^{2} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}} \] Input:

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

Output:

-1/6*(3*a^2*log(sqrt(-a^2*x^2 + 1) + 1)/c^2 - 3*a^2*log(sqrt(-a^2*x^2 + 1) 
 - 1)/c^2 + 2*(3*(a^2*x^2 - 1)*a^2 - a^2)/((-a^2*x^2 + 1)^(3/2)*c^2))/a + 
1/3*(8*a^4*x^4 - 12*a^2*x^2 + 3)/((a^2*c^2*x^3 - c^2*x)*sqrt(a*x + 1)*sqrt 
(-a*x + 1))
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.73 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {a^3\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^2\,x}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {17\,a^2\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2} \] Input:

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

Output:

(a^3*(1 - a^2*x^2)^(1/2))/(6*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) - (1 - 
 a^2*x^2)^(1/2)/(c^2*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^2 + (a^2*( 
1 - a^2*x^2)^(1/2))/(4*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) + (c^2*(-a^2)^(1/2 
))/a)) + (17*a^2*(1 - a^2*x^2)^(1/2))/(12*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) 
 - (c^2*(-a^2)^(1/2))/a))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.30 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {12 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}-12 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x -11 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+11 \sqrt {-a^{2} x^{2}+1}\, a x +32 a^{3} x^{3}-20 a^{2} x^{2}-28 a x +12}{12 \sqrt {-a^{2} x^{2}+1}\, c^{2} x \left (a x -1\right )} \] Input:

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

Output:

(12*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**2*x**2 - 12*sqrt( - a* 
*2*x**2 + 1)*log(tan(asin(a*x)/2))*a*x - 11*sqrt( - a**2*x**2 + 1)*a**2*x* 
*2 + 11*sqrt( - a**2*x**2 + 1)*a*x + 32*a**3*x**3 - 20*a**2*x**2 - 28*a*x 
+ 12)/(12*sqrt( - a**2*x**2 + 1)*c**2*x*(a*x - 1))