\(\int \frac {e^{\text {arctanh}(a x)}}{(c-\frac {c}{a x})^2} \, dx\) [477]

Optimal result

Integrand size = 20, antiderivative size = 94 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {4 (1+a x)}{a c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac {3 \arcsin (a x)}{a c^2} \] Output:

(-4*a*x-4)/a/c^2/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/c^2+1/3*(-a^2*x^2 
+1)^(3/2)/a/c^2/(-a*x+1)^3+3*arcsin(a*x)/a/c^2
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\frac {\sqrt {1+a x} \left (-14+19 a x-3 a^2 x^2\right )}{(1-a x)^{3/2}}-18 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{3 a c^2} \] Input:

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

Output:

((Sqrt[1 + a*x]*(-14 + 19*a*x - 3*a^2*x^2))/(1 - a*x)^(3/2) - 18*ArcSin[Sq 
rt[1 - a*x]/Sqrt[2]])/(3*a*c^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6681, 6678, 581, 25, 671, 463, 25, 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.

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

\(\Big \downarrow \) 6681

\(\displaystyle \frac {a^2 \int \frac {e^{\text {arctanh}(a x)} x^2}{(1-a x)^2}dx}{c^2}\)

\(\Big \downarrow \) 6678

\(\displaystyle \frac {a^2 \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{c^2}\)

\(\Big \downarrow \) 581

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 463

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 223

\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {3 \left (\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}-\frac {\arcsin (a x)}{a}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a (1-a x)^3}}{a^2}\right )}{c^2}\)

Input:

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

Output:

(a^2*((1 - a^2*x^2)^(3/2)/(a^3*(1 - a*x)^2) - (-1/3*(1 - a^2*x^2)^(3/2)/(a 
*(1 - a*x)^3) + 3*((2*Sqrt[1 - a^2*x^2])/(a*(1 - a*x)) - ArcSin[a*x]/a))/a 
^2))/c^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 463

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x 
))), x] - Simp[d^(2*n + 2)/b^(n + 1)   Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ 
(2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F 
reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, 
-3/2]
 

rule 581

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

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

rule 6678

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[c^n   Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - 
 a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 
0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 
, 0]) && IntegerQ[2*p]
 

rule 6681

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.60

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\)	\(150\)
default	\(\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{4}}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}\right )}{c^{2}}\)	\(182\)

Input:

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

Output:

1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^2+(3/a^2/(a^2)^(1/2)*arctan((a^2)^(1/ 
2)*x/(-a^2*x^2+1)^(1/2))+2/3/a^5/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1 
/2)+13/3/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))*a^2/c^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \] Input:

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

Output:

-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 
 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^ 
3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^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 [B] (verification not implemented)

Time = 15.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.49 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )} \] Input:

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

Output:

(2*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) + (3*a 
sinh(x*(-a^2)^(1/2)))/(c^2*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c^2) - ( 
13*(1 - a^2*x^2)^(1/2))/(3*(-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 = 181, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+9 \mathit {asin} \left (a x \right ) a^{2} x^{2}-18 \mathit {asin} \left (a x \right ) a x +9 \mathit {asin} \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+11 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}+3 a^{3} x^{3}-24 a^{2} x^{2}+11 a x +6}{3 a \,c^{2} \left (\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-2 a x +1\right )} \] Input:

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

Output:

(9*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 9*sqrt( - a**2*x**2 + 1)*asin(a* 
x) + 9*asin(a*x)*a**2*x**2 - 18*asin(a*x)*a*x + 9*asin(a*x) - 3*sqrt( - a* 
*2*x**2 + 1)*a**2*x**2 + 11*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x* 
*2 + 1) + 3*a**3*x**3 - 24*a**2*x**2 + 11*a*x + 6)/(3*a*c**2*(sqrt( - a**2 
*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1) + a**2*x**2 - 2*a*x + 1))