\(\int \frac {e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\) [619]

Optimal result

Integrand size = 25, antiderivative size = 86 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}+\frac {2 \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x}+\sqrt {a} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{\sqrt {1-a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6684, 6678, 516, 57, 63, 222}

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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 516

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 63

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

\(\Big \downarrow \) 222

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 516

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free 
Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || 
(GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[n]))
 

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 6684

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a x +2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, \sqrt {a}}\)	\(90\)
risch	\(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {a \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(163\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.74 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\frac {2 \sqrt {c}\, \left (-\mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x +\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a x} \] Input:

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

Output:

(2*sqrt(c)*( - atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a*x + sqr 
t(x)*sqrt(a)*sqrt(a*x + 1)*i))/(a*x)