\(\int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\) [535]

Optimal result

Integrand size = 27, antiderivative size = 152 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {152 a^3 c \sqrt {1-\frac {1}{a^2 x^2}}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {38}{105} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {4 a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{35 c}-\frac {2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}}{7 c^2} \] Output:

-152/105*a^3*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)-38/105*a^3*(1-1/a^2/x^2 
)^(1/2)*(c-c/a/x)^(1/2)+4/35*a^3*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/c-2/7 
*a^3*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(5/2)/c^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (-15+39 a x-52 a^2 x^2+104 a^3 x^3\right )}{105 x^2 (-1+a x)} \] Input:

Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^4),x]
 

Output:

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-15 + 39*a*x - 52*a^2*x^2 + 
 104*a^3*x^3))/(105*x^2*(-1 + a*x))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6733, 574, 581, 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.

\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-\coth ^{-1}(a x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 574

\(\displaystyle -\frac {\frac {13}{7} c \int \frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}-\frac {2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{7 x^3 \sqrt {c-\frac {c}{a x}}}}{c}\)

\(\Big \downarrow \) 581

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 672

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

\(\Big \downarrow \) 458

\(\displaystyle -\frac {\frac {13}{7} c \left (\frac {1}{5} a \left (\frac {14 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}-\frac {4}{3} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}\right )+\frac {2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}\right )-\frac {2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{7 x^3 \sqrt {c-\frac {c}{a x}}}}{c}\)

Input:

Int[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^4),x]
 

Output:

-(((13*c*((a*((14*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/(3*Sqrt[c - c/(a*x)]) - (4* 
a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/3))/5 + (2*a^3*Sqrt[1 - 1/(a^ 
2*x^2)]*(c - c/(a*x))^(3/2))/(5*c)))/7 - (2*c^2*Sqrt[1 - 1/(a^2*x^2)])/(7* 
Sqrt[c - c/(a*x)]*x^3))/c)
 

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 574

Int[((e_.)*(x_))^(n_)*((c_) + (d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^2*(e*x)^(n + 1)*(c + d*x)^(m - 2)*((a + b*x^2)^(p + 1)/ 
(b*e*(n + p + 2))), x] + Simp[c*((2*n + p + 3)/(n + p + 2))   Int[(e*x)^n*( 
c + d*x)^(m - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x 
] && EqQ[b*c^2 + a*d^2, 0] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1] && IntegerQ 
[2*p]
 

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 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 = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.45

method	result	size
orering	\(-\frac {2 \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{105 x^{3} \left (a x -1\right )}\)	\(68\)
gosper	\(-\frac {2 \left (a x +1\right ) \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{105 \left (a x -1\right ) x^{3}}\)	\(70\)
default	\(-\frac {2 \left (a x +1\right ) \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{105 \left (a x -1\right ) x^{3}}\)	\(70\)
risch	\(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (104 a^{4} x^{4}+52 a^{3} x^{3}-13 a^{2} x^{2}+24 a x -15\right )}{105 \left (a x -1\right ) x^{3}}\)	\(73\)

Input:

int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-2/105*(104*a^3*x^3-52*a^2*x^2+39*a*x-15)*(a*x+1)/x^3/(a*x-1)*(c-c/a/x)^(1 
/2)*((a*x-1)/(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2 \, {\left (104 \, a^{4} x^{4} + 52 \, a^{3} x^{3} - 13 \, a^{2} x^{2} + 24 \, a x - 15\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="fricas 
")
 

Output:

-2/105*(104*a^4*x^4 + 52*a^3*x^3 - 13*a^2*x^2 + 24*a*x - 15)*sqrt((a*x - 1 
)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^4 - x^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\text {Timed out} \] Input:

integrate((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(1/2)/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{4}} \,d x } \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="maxima 
")
 

Output:

integrate(sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1))/x^4, x)
 

Giac [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{4}} \,d x } \] Input:

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^4,x, algorithm="giac")
 

Output:

integrate(sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1))/x^4, x)
 

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (104\,a^3\,x^3+156\,a^2\,x^2+143\,a\,x+167\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3}-\frac {304\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )} \] Input:

int(((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/x^4,x)
 

Output:

- (2*((a*x - 1)/(a*x + 1))^(1/2)*(143*a*x + 156*a^2*x^2 + 104*a^3*x^3 + 16 
7)*((c*(a*x - 1))/(a*x))^(1/2))/(105*x^3) - (304*((a*x - 1)/(a*x + 1))^(1/ 
2)*((c*(a*x - 1))/(a*x))^(1/2))/(105*x^3*(a*x - 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.53 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c}\, \left (-104 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+52 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-39 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +15 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+104 a^{4} x^{4}\right )}{105 a \,x^{4}} \] Input:

int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^4,x)
 

Output:

(2*sqrt(c)*( - 104*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 + 52*sqrt(x)*sq 
rt(a)*sqrt(a*x + 1)*a**2*x**2 - 39*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x + 15* 
sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 104*a**4*x**4))/(105*a*x**4)