\(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [553]

Optimal result

Integrand size = 27, antiderivative size = 228 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {928 a^4 c \sqrt {1-\frac {1}{a^2 x^2}}}{45 \sqrt {c-\frac {c}{a x}}}-\frac {232}{45} a^4 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}-\frac {29 a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{15 c}-\frac {2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}}{9 c^2}-\frac {a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{7/2}}{9 c^3} \] Output:

-928/45*a^4*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)-232/45*a^4*(1-1/a^2/x^2) 
^(1/2)*(c-c/a/x)^(1/2)-29/15*a^4*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/c-2/9 
*a^4*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(5/2)/c^2-a^4*(c-c/a/x)^(7/2)/c^3/(1-1/ 
a^2/x^2)^(1/2)-2/9*a^4*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(7/2)/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.38 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (5-20 a x+41 a^2 x^2-82 a^3 x^3+328 a^4 x^4+656 a^5 x^5\right )}{45 x^3 \left (-1+a^2 x^2\right )} \] Input:

Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5),x]
 

Output:

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(5 - 20*a*x + 41*a^2*x^2 - 8 
2*a^3*x^3 + 328*a^4*x^4 + 656*a^5*x^5))/(45*x^3*(-1 + a^2*x^2))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6733, 581, 27, 2166, 27, 672, 459, 459, 458}

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.

\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx\)

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 581

\(\displaystyle -\frac {-\frac {2 a^3 \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {13 c^3}{a x}-\frac {7 c^3}{a^2 x^2}+11 c^3\right )}{2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {a^3 \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {13 c^3}{a x}-\frac {7 c^3}{a^2 x^2}+11 c^3\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {-\frac {a^3 \left (-c \int \frac {c^3 \left (97 a+\frac {14}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {a^3 \left (-\frac {c^4 \int \frac {\left (97 a+\frac {14}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 672

\(\displaystyle -\frac {-\frac {a^3 \left (-\frac {c^4 \left (87 a \int \frac {\left (c-\frac {c}{a x}\right )^{5/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a}-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 459

\(\displaystyle -\frac {-\frac {a^3 \left (-\frac {c^4 \left (87 a \left (\frac {8}{5} c \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2}{5} a c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}\right )-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a}-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 459

\(\displaystyle -\frac {-\frac {a^3 \left (-\frac {c^4 \left (87 a \left (\frac {8}{5} c \left (\frac {4}{3} c \int \frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2}{3} a c \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}\right )+\frac {2}{5} a c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}\right )-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a}-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{9 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\)

\(\Big \downarrow \) 458

\(\displaystyle -\frac {-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{11/2}}{9 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {a^3 \left (-\frac {17 a c^3 \left (c-\frac {c}{a x}\right )^{7/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^4 \left (87 a \left (\frac {8}{5} c \left (\frac {8 a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}+\frac {2}{3} a c \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}\right )+\frac {2}{5} a c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}\right )-4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a}\right )}{9 c^3}}{c^3}\)

Input:

Int[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5),x]
 

Output:

-((-1/9*(a^3*(-1/2*(c^4*(87*a*((8*c*((8*a*c^2*Sqrt[1 - 1/(a^2*x^2)])/(3*Sq 
rt[c - c/(a*x)]) + (2*a*c*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/3))/5 + 
 (2*a*c*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/5) - 4*a^2*Sqrt[1 - 1/( 
a^2*x^2)]*(c - c/(a*x))^(5/2)))/a - (17*a*c^3*(c - c/(a*x))^(7/2))/Sqrt[1 
- 1/(a^2*x^2)]))/c^3 - (2*a^4*(c - c/(a*x))^(11/2))/(9*c^2*Sqrt[1 - 1/(a^2 
*x^2)]))/c^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 458

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

rule 459

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
(Simplify[n + p]/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif 
y[n + p], 0]
 

rule 581

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

rule 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^ 
2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 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 = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.37

method	result	size
orering	\(-\frac {2 \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}}\)	\(84\)
gosper	\(-\frac {2 \left (a x +1\right ) \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}}\)	\(86\)
default	\(-\frac {2 \left (a x +1\right ) \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}}\)	\(86\)
risch	\(-\frac {2 \left (476 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{45 x^{4} \left (a x -1\right )}-\frac {8 x \,a^{5} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\)	\(125\)

Input:

int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-2/45*(656*a^5*x^5+328*a^4*x^4-82*a^3*x^3+41*a^2*x^2-20*a*x+5)*(a*x+1)/x^4 
/(a*x-1)^2*(c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 \, {\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{45 \, {\left (a x^{5} - x^{4}\right )}} \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="fricas 
")
 

Output:

-2/45*(656*a^5*x^5 + 328*a^4*x^4 - 82*a^3*x^3 + 41*a^2*x^2 - 20*a*x + 5)*s 
qrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^5 - x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Timed out} \] Input:

integrate((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="maxima 
")
 

Output:

integrate(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)/x^5, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,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 [B] (verification not implemented)

Time = 13.73 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}\,\left (656\,a^4\,x^4+984\,a^3\,x^3+902\,a^2\,x^2+943\,a\,x+923\right )}{45\,x^4}-\frac {1856\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{45\,x^4\,\left (a\,x-1\right )} \] Input:

int(((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^5,x)
 

Output:

- (2*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2)*(943*a*x + 90 
2*a^2*x^2 + 984*a^3*x^3 + 656*a^4*x^4 + 923))/(45*x^4) - (1856*((a*x - 1)/ 
(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2))/(45*x^4*(a*x - 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (656 \sqrt {a x +1}\, a^{5} x^{5}-656 \sqrt {x}\, \sqrt {a}\, a^{5} x^{5}-328 \sqrt {x}\, \sqrt {a}\, a^{4} x^{4}+82 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}-41 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+20 \sqrt {x}\, \sqrt {a}\, a x -5 \sqrt {x}\, \sqrt {a}\right )}{45 \sqrt {a x +1}\, a \,x^{5}} \] Input:

int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x)
 

Output:

(2*sqrt(c)*(656*sqrt(a*x + 1)*a**5*x**5 - 656*sqrt(x)*sqrt(a)*a**5*x**5 - 
328*sqrt(x)*sqrt(a)*a**4*x**4 + 82*sqrt(x)*sqrt(a)*a**3*x**3 - 41*sqrt(x)* 
sqrt(a)*a**2*x**2 + 20*sqrt(x)*sqrt(a)*a*x - 5*sqrt(x)*sqrt(a)))/(45*sqrt( 
a*x + 1)*a*x**5)