\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a x})^2 \, dx\) [520]

Optimal result

Integrand size = 22, antiderivative size = 82 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {3 c^2 \arcsin (a x)}{a}+\frac {3 c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (-\frac {(-1+a x)^2 (1+a x)}{x \sqrt {1-a^2 x^2}}+3 a \arcsin (a x)+3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6681, 6678, 540, 2340, 27, 538, 223, 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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 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 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 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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 = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.23

method	result	size
risch	\(\frac {\left (a^{2} x^{2}-1\right ) c^{2}}{x \sqrt {-a^{2} x^{2}+1}\, a^{2}}+\frac {\left (a \sqrt {-a^{2} x^{2}+1}+3 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{2}}{a^{2}}\)	\(101\)
default	\(\frac {c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )-3 a \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+4 a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\right )}{a^{2}}\)	\(170\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {6 \, a c^{2} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a c^{2} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x - {\left (a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.70 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {a^{2} c^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} + \frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {3 \, c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x {\left | a \right |}} \] Input:

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

Output:

1/2*a^2*c^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) + 3*c^2*arcsin(a*x) 
*sgn(a)/abs(a) + 3*c^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^ 
2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c^2/a - 1/2*(sqrt(-a^2*x^2 + 1)*abs 
(a) + a)*c^2/(a^2*x*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{a} \] Input:

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

Output:

(3*c^2*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (c^2*atan((1 - a^2*x^2)^(1/2) 
*1i)*3i)/a + (c^2*(1 - a^2*x^2)^(1/2))/a - (c^2*(1 - a^2*x^2)^(1/2))/(a^2* 
x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (3 \mathit {asin} \left (a x \right ) a x +\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}-3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x -a x \right )}{a^{2} x} \] Input:

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

Output:

(c**2*(3*asin(a*x)*a*x + sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 
1) - 3*log(tan(asin(a*x)/2))*a*x - a*x))/(a**2*x)