\(\int \frac {e^{-3 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^2} \, dx\) [523]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {\arcsin (a x)}{a c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {\frac {\sqrt {1-a x} (2+a x)}{\sqrt {1+a x}}+\arcsin (a x)}{a c^2} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6681, 6678, 527, 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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 527

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 223

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

Input:

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^2),x]
 

Output:

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

Defintions of rubi rules used

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 527

Int[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_S 
ymbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + 
b*x^2])), x] + Simp[1/(b*d^(m - 2))   Int[(1/Sqrt[a + b*x^2])*ExpandToSum[( 
2^(n - 1)*c^(m + n - 1) - d^m*x^m*(c + d*x)^(n - 1))/(c - d*x), x], x], x] 
/; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && EqQ[b*c^2 + a*d^2, 
 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 = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\)	\(102\)
default	\(\frac {a^{2} \left (\frac {-\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}-3 a \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{8 a^{4}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{4 a^{5}}-\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{4 a^{4}}-\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{16 a^{3}}+\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{16 a^{3}}\right )}{c^{2}}\)	\(769\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {2 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2}{a^{2} c^{2} x + a c^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} + \frac {2}{c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \] Input:

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

Output:

-arcsin(a*x)*sgn(a)/(c^2*abs(a)) - sqrt(-a^2*x^2 + 1)/(a*c^2) + 2/(c^2*((s 
qrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-\sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+\mathit {asin} \left (a x \right ) a x +\mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}+a x -2}{a \,c^{2} \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:

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

Output:

( - sqrt( - a**2*x**2 + 1)*asin(a*x) + asin(a*x)*a*x + asin(a*x) + sqrt( - 
 a**2*x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1) + a**2*x**2 + a*x - 2)/(a*c 
**2*(sqrt( - a**2*x**2 + 1) - a*x - 1))