\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\) [540]

Optimal result

Integrand size = 27, antiderivative size = 86 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=2 \sqrt {c-\frac {c}{a x}}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=2 \sqrt {c-\frac {c}{a x}}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

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

Output:

2*Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 4*Sqr 
t[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6717, 6683, 1070, 281, 948, 95, 27, 174, 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.

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

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6683

\(\displaystyle -\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x (a x+1)}dx\)

\(\Big \downarrow \) 1070

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 95

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \left (\frac {c^2 \left (\frac {8 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}\right )}{a}-\frac {2 c \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \left (\frac {c^2 \left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a}-\frac {2 c \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\)

Input:

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

Output:

-((a*((-2*c*Sqrt[c - c/(a*x)])/a + (c^2*((-2*ArcTanh[Sqrt[c - c/(a*x)]/Sqr 
t[c]])/Sqrt[c] + (4*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/ 
Sqrt[c]))/a))/c)
 

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 95

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p - 1)/(b*d*(p - 1))), x] + Simp[1/(b*d)   Int[(b 
*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*((e + f*x)^(p - 2)/((a + 
b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 948

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

rule 1070

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^ 
(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(m + n*(p + r))*( 
b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, 
 n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

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

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(69)=138\).

Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92

method	result	size
risch	\(2 \sqrt {\frac {c \left (a x -1\right )}{a x}}+\frac {\left (\frac {a \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{a x -1}\)	\(165\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-4 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x^{2}+2 \sqrt {x \left (a x -1\right )}\, x^{2} a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}+2 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a \,x^{2}-2 \sqrt {a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{2}-3 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a \,x^{2}\right )}{x \sqrt {x \left (a x -1\right )}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\)	\(228\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.56 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [2 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, \sqrt {\frac {a c x - c}{a x}}, 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 2 \, \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 2 \, \sqrt {\frac {a c x - c}{a x}}\right ] \] Input:

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

Output:

[2*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3* 
a*c*x + c)/(a*x + 1)) + sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - 
 c)/(a*x)) + c) + 2*sqrt((a*c*x - c)/(a*x)), 4*sqrt(2)*sqrt(-c)*arctan(sqr 
t(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) - 2*sqrt(-c)*arctan 
(a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + 2*sqrt((a*c*x - c)/(a 
*x))]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (70) = 140\).

Time = 4.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.64 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\begin {cases} - \frac {2 a \left (\frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {c - \frac {c}{a x}}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {c \sqrt {c - \frac {c}{a x}}}{a}\right )}{c} & \text {for}\: \frac {c}{a} \neq 0 \\- \frac {3 a \sqrt {c} \left (\frac {\log {\left (- \frac {2}{x} \right )}}{a} - \frac {\log {\left (2 a + \frac {2}{x} \right )}}{a}\right )}{2} + \frac {\sqrt {c} \log {\left (\frac {a}{x} + \frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-2*a*(c**2*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 2*sq 
rt(2)*c**2*atan(sqrt(2)*sqrt(c - c/(a*x))/(2*sqrt(-c)))/(a*sqrt(-c)) - c*s 
qrt(c - c/(a*x))/a)/c, Ne(c/a, 0)), (-3*a*sqrt(c)*(log(-2/x)/a - log(2*a + 
 2/x)/a)/2 + sqrt(c)*log(a/x + x**(-2))/2, True))
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x,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 [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.40 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-\sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a x -\sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a x +\sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a x +\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a x +a x \right )}{a x} \] Input:

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

Output:

(2*sqrt(c)*(sqrt(x)*sqrt(a)*sqrt(a*x - 1) - sqrt(2)*log(sqrt(a*x - 1) + sq 
rt(x)*sqrt(a) - sqrt(2)*i + i)*a*x - sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*s 
qrt(a) + sqrt(2)*i - i)*a*x + sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) 
+ 2*sqrt(2) + 2*a*x + 2)*a*x + log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a*x + 
a*x))/(a*x)