\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [544]

Optimal result

Integrand size = 27, antiderivative size = 163 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

4*a^4*(c-c/a/x)^(1/2)+2/3*a^4*(c-c/a/x)^(3/2)/c+2/5*a^4*(c-c/a/x)^(5/2)/c^ 
2-2/7*a^4*(c-c/a/x)^(7/2)/c^3+2/9*a^4*(c-c/a/x)^(9/2)/c^4-4*2^(1/2)*a^4*c^ 
(1/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (35-95 a x+138 a^2 x^2-236 a^3 x^3+788 a^4 x^4\right )}{315 x^4}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^5),x]
 

Output:

(2*Sqrt[c - c/(a*x)]*(35 - 95*a*x + 138*a^2*x^2 - 236*a^3*x^3 + 788*a^4*x^ 
4))/(315*x^4) - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*S 
qrt[c])]
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6717, 6683, 1070, 281, 948, 99, 2009}

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.

\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-2 \coth ^{-1}(a x)}}{x^5} \, dx\)

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1070

\(\displaystyle -\int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}}}{\left (a+\frac {1}{x}\right ) x^5}dx\)

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 99

\(\displaystyle -\frac {a \int \left (\frac {a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{c^2}-\frac {a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{c}+a^2 \left (c-\frac {c}{a x}\right )^{3/2}-\frac {a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{a+\frac {1}{x}}\right )d\frac {1}{x}}{c}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (4 \sqrt {2} a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^3}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^2}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c}-\frac {2}{3} a^3 \left (c-\frac {c}{a x}\right )^{3/2}-4 a^3 c \sqrt {c-\frac {c}{a x}}\right )}{c}\)

Input:

Int[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^5),x]
 

Output:

-((a*(-4*a^3*c*Sqrt[c - c/(a*x)] - (2*a^3*(c - c/(a*x))^(3/2))/3 - (2*a^3* 
(c - c/(a*x))^(5/2))/(5*c) + (2*a^3*(c - c/(a*x))^(7/2))/(7*c^2) - (2*a^3* 
(c - c/(a*x))^(9/2))/(9*c^3) + 4*Sqrt[2]*a^3*c^(3/2)*ArcTanh[Sqrt[c - c/(a 
*x)]/(Sqrt[2]*Sqrt[c])]))/c)
 

Defintions of rubi rules used

rule 99

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 [A] (verified)

Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {2 \left (788 a^{5} x^{5}-1024 a^{4} x^{4}+374 a^{3} x^{3}-233 a^{2} x^{2}+130 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \left (a x -1\right )}+\frac {2 a^{4} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{\sqrt {c}\, \left (a x -1\right )}\)	\(168\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-1890 \sqrt {a \,x^{2}-x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}+630 a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, x^{6}+1260 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{4}+945 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-630 a^{\frac {9}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{6}-945 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-316 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{3}+156 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{2}-120 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x +70 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{315 x^{5} \sqrt {x \left (a x -1\right )}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\)	\(326\)

Input:

int((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x^5,x,method=_RETURNVERBOSE)
 

Output:

2/315*(788*a^5*x^5-1024*a^4*x^4+374*a^3*x^3-233*a^2*x^2+130*a*x-35)/x^4/(a 
*x-1)*(c*(a*x-1)/a/x)^(1/2)+2*a^4*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 = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}, \frac {2 \, {\left (630 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}\right ] \] Input:

integrate((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="fricas")
 

Output:

[2/315*(315*sqrt(2)*a^4*sqrt(c)*x^4*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x 
 - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + (788*a^4*x^4 - 236*a^3*x^3 + 138* 
a^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4, 2/315*(630*sqrt(2)*a^ 
4*sqrt(-c)*x^4*arctan(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x 
- c)) + (788*a^4*x^4 - 236*a^3*x^3 + 138*a^2*x^2 - 95*a*x + 35)*sqrt((a*c* 
x - c)/(a*x)))/x^4]
 

Sympy [F]

\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \] Input:

integrate((c-c/a/x)**(1/2)*(a*x-1)/(a*x+1)/x**5,x)
 

Output:

Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x - 1)/(x**5*(a*x + 1)), x)
 

Maxima [F]

\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{5}} \,d x } \] Input:

integrate((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="maxima")
 

Output:

integrate((a*x - 1)*sqrt(c - c/(a*x))/((a*x + 1)*x^5), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (136) = 272\).

Time = 0.37 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.66 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {4 \, \sqrt {2} a^{5} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a^{9} c - 1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{8} c^{\frac {3}{2}} {\left | a \right |} + 2100 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{9} c^{2} - 3150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{8} c^{\frac {5}{2}} {\left | a \right |} + 3528 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{9} c^{3} - 2625 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{8} c^{\frac {7}{2}} {\left | a \right |} + 1215 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{9} c^{4} - 315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{8} c^{\frac {9}{2}} {\left | a \right |} + 35 \, a^{9} c^{5}\right )}}{315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right )} \] Input:

integrate((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="giac")
 

Output:

-4*sqrt(2)*a^5*c*arctan(-1/2*sqrt(2)*((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a* 
c*x))*a + sqrt(c)*abs(a))/(a*sqrt(-c)))/(sqrt(-c)*abs(a)*sgn(x)) + 2/315*( 
1260*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^8*a^9*c - 1260*(sqrt(a^2*c) 
*x - sqrt(a^2*c*x^2 - a*c*x))^7*a^8*c^(3/2)*abs(a) + 2100*(sqrt(a^2*c)*x - 
 sqrt(a^2*c*x^2 - a*c*x))^6*a^9*c^2 - 3150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 
 - a*c*x))^5*a^8*c^(5/2)*abs(a) + 3528*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a 
*c*x))^4*a^9*c^3 - 2625*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^3*a^8*c^ 
(7/2)*abs(a) + 1215*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a^9*c^4 - 
315*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a^8*c^(9/2)*abs(a) + 35*a^9* 
c^5)/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^9*a^4*abs(a)*sgn(x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^5\,\left (a\,x+1\right )} \,d x \] Input:

int(((c - c/(a*x))^(1/2)*(a*x - 1))/(x^5*(a*x + 1)),x)
 

Output:

int(((c - c/(a*x))^(1/2)*(a*x - 1))/(x^5*(a*x + 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (788 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{4} x^{4}-236 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}+138 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}-95 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x +35 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{5} x^{5}-315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{5} x^{5}+315 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{5} x^{5}-508 a^{5} x^{5}\right )}{315 a \,x^{5}} \] Input:

int((c-c/a/x)^(1/2)*(a*x-1)/(a*x+1)/x^5,x)
 

Output:

(2*sqrt(c)*(788*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**4*x**4 - 236*sqrt(x)*sqrt 
(a)*sqrt(a*x - 1)*a**3*x**3 + 138*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**2*x**2 
- 95*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x + 35*sqrt(x)*sqrt(a)*sqrt(a*x - 1) 
- 315*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt(2)*i + i)*a**5*x* 
*5 - 315*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a**5 
*x**5 + 315*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a* 
x + 2)*a**5*x**5 - 508*a**5*x**5))/(315*a*x**5)