\(\int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [510]

Optimal result

Integrand size = 25, antiderivative size = 152 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {26 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {46 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {10}{21} a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{9 c} \] Output:

-26/315*a^4*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)+46/105*a^4*c*(1-1/a^2/ 
x^2)^(3/2)/(c-c/a/x)^(1/2)-10/21*a^4*(1-1/a^2/x^2)^(3/2)*(c-c/a/x)^(1/2)+2 
/9*a^4*(1-1/a^2/x^2)^(3/2)*(c-c/a/x)^(3/2)/c
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.49 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (-35-5 a x+6 a^2 x^2-8 a^3 x^3+16 a^4 x^4\right )}{315 x^3 (-1+a x)} \] Input:

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

Output:

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-35 - 5*a*x + 6*a^2*x^2 - 8* 
a^3*x^3 + 16*a^4*x^4))/(315*x^3*(-1 + a*x))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6733, 581, 27, 2170, 27, 672, 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.

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

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 581

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle -c \left (-\frac {a^3 \left (-\frac {2 a^4 \int -\frac {c^5 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {23}{x}\right )}{2 a^5 \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{7 c^2}-\frac {10}{7} a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}\right )}{3 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{9 c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -c \left (-\frac {a^3 \left (\frac {c^3 \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {23}{x}\right )}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{7 a}-\frac {10}{7} a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}\right )}{3 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{9 c^2}\right )\)

\(\Big \downarrow \) 672

\(\displaystyle -c \left (-\frac {a^3 \left (\frac {c^3 \left (\frac {46 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}-\frac {13}{5} a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\right )}{7 a}-\frac {10}{7} a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}\right )}{3 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{9 c^2}\right )\)

\(\Big \downarrow \) 458

\(\displaystyle -c \left (-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{9 c^2}-\frac {a^3 \left (\frac {c^3 \left (\frac {46 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}-\frac {26 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{7 a}-\frac {10}{7} a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}\right )}{3 c^3}\right )\)

Input:

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

Output:

-(c*(-1/3*(a^3*((c^3*((-26*a^2*c*(1 - 1/(a^2*x^2))^(3/2))/(15*(c - c/(a*x) 
)^(3/2)) + (46*a^2*(1 - 1/(a^2*x^2))^(3/2))/(5*Sqrt[c - c/(a*x)])))/(7*a) 
- (10*a*c^2*(1 - 1/(a^2*x^2))^(3/2)*Sqrt[c - c/(a*x)])/7))/c^3 - (2*a^4*(1 
 - 1/(a^2*x^2))^(3/2)*(c - c/(a*x))^(3/2))/(9*c^2)))
 

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 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 2170

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + 
 p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 
 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 
0] &&  !IGtQ[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 = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.40

method	result	size
orering	\(\frac {2 \left (16 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -35\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(61\)
gosper	\(\frac {2 \left (a x +1\right ) \left (16 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(63\)
default	\(\frac {2 \left (a x +1\right ) \left (16 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(63\)
risch	\(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (16 a^{5} x^{5}+8 a^{4} x^{4}-2 a^{3} x^{3}+a^{2} x^{2}-40 a x -35\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x^{4}}\)	\(80\)

Input:

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

Output:

2/315*(16*a^3*x^3-24*a^2*x^2+30*a*x-35)/x^4*(a*x+1)/((a*x-1)/(a*x+1))^(1/2 
)*(c-c/a/x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \, {\left (16 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} - 40 \, a x - 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}} \] Input:

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

Output:

2/315*(16*a^5*x^5 + 8*a^4*x^4 - 2*a^3*x^3 + a^2*x^2 - 40*a*x - 35)*sqrt((a 
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^5 - x^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 13.71 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}\,\left (16\,a^4\,x^4+24\,a^3\,x^3+22\,a^2\,x^2+23\,a\,x-17\right )}{315\,x^4}-\frac {104\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{315\,x^4\,\left (a\,x-1\right )} \] Input:

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

Output:

(2*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2)*(23*a*x + 22*a^ 
2*x^2 + 24*a^3*x^3 + 16*a^4*x^4 - 17))/(315*x^4) - (104*((a*x - 1)/(a*x + 
1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2))/(315*x^4*(a*x - 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (16 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{4} x^{4}-8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-5 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -35 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-16 a^{5} x^{5}\right )}{315 a \,x^{5}} \] Input:

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

Output:

(2*sqrt(c)*(16*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**4*x**4 - 8*sqrt(x)*sqrt(a) 
*sqrt(a*x + 1)*a**3*x**3 + 6*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 5*s 
qrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 35*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 16*a 
**5*x**5))/(315*a*x**5)