\(\int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) [623]

Optimal result

Integrand size = 27, antiderivative size = 147 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.63 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x} (1+7 a x)+6 \sqrt {2} a^{3/2} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{3 x \sqrt {1-a x}} \] Input:

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

Output:

(2*Sqrt[c - c/(a*x)]*(-(Sqrt[1 + a*x]*(1 + 7*a*x)) + 6*Sqrt[2]*a^(3/2)*x^( 
3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(3*x*Sqrt[1 - a*x] 
)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6684, 6679, 105, 105, 104, 219}

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.

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

\(\Big \downarrow \) 6684

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{5/2}}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 6679

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(a x+1)^{3/2}}{x^{5/2} (1-a x)}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6679

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

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 = 0.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 x \sqrt {-x \left (a x +1\right )}\, a \sqrt {2}\, \sqrt {-\frac {1}{a}}+6 a \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x^{2}+\sqrt {-x \left (a x +1\right )}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\right ) \sqrt {2}}{3 x \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\)	\(150\)
risch	\(-\frac {2 \left (7 a^{2} x^{2}+8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {4 a \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-2 c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(206\)

Input:

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

Output:

1/3*(c*(a*x-1)/a/x)^(1/2)/x*(-a^2*x^2+1)^(1/2)*(7*x*(-x*(a*x+1))^(1/2)*a*2 
^(1/2)*(-1/a)^(1/2)+6*a*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+(-x*(a*x+1))^(1/2)*2^(1/2)*(-1/a)^(1/2))*2^(1/2)/(a*x- 
1)/(-x*(a*x+1))^(1/2)/(-1/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.10 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, -\frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \] Input:

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

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^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt( 
(a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*sqrt(-a^2*x 
^2 + 1)*(7*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)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sq 
rt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - sqrt(-a^2*x^2 + 1)*(7 
*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.48 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c}\, \left (-6 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a^{2} x^{2}+7 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{3 a \,x^{2}} \] Input:

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

Output:

(2*sqrt(c)*( - 6*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a 
*x + 1))*a**2*x**2 + 7*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x + sqrt(x)*sqrt( 
a)*sqrt(a*x + 1)*i))/(3*a*x**2)