\(\int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) [525]

Optimal result

Integrand size = 27, antiderivative size = 125 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.24 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 a \left (\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} (1+7 a x)+3 \sqrt {2} \sqrt {c} (-1+a x) \log \left ((-1+a x)^2\right )-3 \sqrt {2} \sqrt {c} (-1+a x) \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right )\right )}{-3+3 a x} \] Input:

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

Output:

(2*a*(Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(1 + 7*a*x) + 3*Sqrt[2]*Sqrt 
[c]*(-1 + a*x)*Log[(-1 + a*x)^2] - 3*Sqrt[2]*Sqrt[c]*(-1 + a*x)*Log[2*Sqrt 
[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a* 
x + 3*a^2*x^2)]))/(-3 + 3*a*x)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6733, 466, 466, 471, 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.

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

\(\Big \downarrow \) 6733

\(\displaystyle -c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}\)

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 466

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

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

rule 6733

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {2 \left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-3 a \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{2}+7 x \sqrt {x \left (a x +1\right )}\, a \sqrt {\frac {1}{a}}+\sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}\right )}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\)	\(140\)
risch	\(\frac {2 \left (7 a^{2} x^{2}+8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {2 a \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(180\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.82 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, \frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \] Input:

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

Output:

[1/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 
 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1) 
/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1) 
) + 2*(7*a^2*x^2 + 8*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/( 
a*x)))/(a*x^2 - x), 2/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*arctan(2*sqrt( 
2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a* 
x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + (7*a^2*x^2 + 8*a*x + 1)*sqrt((a*x - 1)/ 
(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/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 [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{x^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.22 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c}\, \left (7 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a^{2} x^{2}-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a^{2} x^{2}-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a^{2} x^{2}+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a^{2} x^{2}-a^{2} x^{2}\right )}{3 a \,x^{2}} \] Input:

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

Output:

(2*sqrt(c)*(7*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x + sqrt(x)*sqrt(a)*sqrt(a*x 
 + 1) + 3*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a**2* 
x**2 - 3*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a**2*x 
**2 - 3*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1)*a**2*x* 
*2 + 3*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1)*a**2*x** 
2 - a**2*x**2))/(3*a*x**2)