\(\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) [639]

Optimal result

Integrand size = 27, antiderivative size = 82 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

4*a*(c-c/a/x)^(1/2)+2/3*a*(c-c/a/x)^(3/2)/c-4*2^(1/2)*a*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 = 69, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} (-1+7 a x)}{3 x}-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

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

Output:

(2*Sqrt[c - c/(a*x)]*(-1 + 7*a*x))/(3*x) - 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqr 
t[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6683, 1070, 281, 946, 60, 60, 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.

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1070

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 946

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[(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] && EqQ[m - n 
+ 1, 0]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(67)=134\).

Time = 0.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.73

method	result	size
risch	\(\frac {2 \left (7 a^{2} x^{2}-8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \left (a x -1\right )}+\frac {2 a \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {c}\, \left (a x -1\right )}\)	\(142\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}-18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}+12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x +9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-6 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{3}-9 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{3 x^{2} \sqrt {x \left (a x -1\right )}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\)	\(254\)

Input:

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

Output:

2/3*(7*a^2*x^2-8*a*x+1)/x*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)+2*a*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))*(c*(a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.01 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} a \sqrt {c} x \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}, \frac {2 \, {\left (6 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}\right ] \] Input:

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

Output:

[2/3*(3*sqrt(2)*a*sqrt(c)*x*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a 
*x)) - 3*a*c*x + c)/(a*x + 1)) + (7*a*x - 1)*sqrt((a*c*x - c)/(a*x)))/x, 2 
/3*(6*sqrt(2)*a*sqrt(-c)*x*arctan(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a 
*x))/(a*c*x - c)) + (7*a*x - 1)*sqrt((a*c*x - c)/(a*x)))/x]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 14.61 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=4\,a\,\sqrt {c-\frac {c}{a\,x}}+\frac {2\,a\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-4\,\sqrt {2}\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}}{2\,\sqrt {c}}\right ) \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.67 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c}\, \left (7 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -\sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{2} x^{2}-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{2} x^{2}+3 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{2} x^{2}-a^{2} x^{2}\right )}{3 a \,x^{2}} \] Input:

int((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/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)*i + i)*a** 
2*x**2 - 3*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a* 
*2*x**2 + 3*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a* 
x + 2)*a**2*x**2 - a**2*x**2))/(3*a*x**2)