\(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^2 \, dx\) [396]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(62)=124\).

Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.55 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (-2-2 a x+2 a^2 x^2+2 a^3 x^3-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {1}{a x}\right )-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6731, 27, 536, 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.

\(\displaystyle \int \left (c-\frac {c}{a x}\right )^2 e^{\coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[2*p]
 

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 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^n   Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ 
x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] 
&& IntegerQ[2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(58)=116\).

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.11

method	result	size
risch	\(\frac {\left (a x -1\right ) c^{2}}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(131\)
default	\(-\frac {\left (a x -1\right ) c^{2} \left (-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(168\)

Input:

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

Output:

(a*x-1)/x*c^2/a^2/((a*x-1)/(a*x+1))^(1/2)+1/a*(((a*x-1)*(a*x+1))^(1/2)+arc 
tan(1/(a^2*x^2-1)^(1/2))-a*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^( 
1/2))*c^2/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (58) = 116\).

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.92 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} + 2 \, a c^{2} x + c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (58) = 116\).

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.02 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-{\left (\frac {4 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (58) = 116\).

Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.21 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, c^{2}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:

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

Output:

-2*c^2*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) + c^2*log(ab 
s(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1 
)*c^2/(a*sgn(a*x + 1)) + 2*c^2/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)*abs 
(a)*sgn(a*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.45 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:

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

Output:

(4*c^2*((a*x - 1)/(a*x + 1))^(1/2))/(a - (a*(a*x - 1)^2)/(a*x + 1)^2) - (2 
*c^2*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (2*c^2*atanh(((a*x - 1)/(a*x + 
 1))^(1/2)))/a
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.66 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (-2 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x +2 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x +\sqrt {a x +1}\, \sqrt {a x -1}\, a x +\sqrt {a x +1}\, \sqrt {a x -1}-2 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +a x \right )}{a^{2} x} \] Input:

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

Output:

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