\(\int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\) [648]

Optimal result

Integrand size = 27, antiderivative size = 169 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {208 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{7 x^3 \sqrt {1-a x}}+\frac {26 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{35 x^2 \sqrt {1-a x}}-\frac {104 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}} \] Output:

208/105*a^3*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a*x+1)^(1/2)-2/7*(c-c/a/x)^(1/ 
2)*(a*x+1)^(1/2)/x^3/(-a*x+1)^(1/2)+26/35*a*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/ 
x^2/(-a*x+1)^(1/2)-104/105*a^2*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/x/(-a*x+1)^(1 
/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x} \left (-15+39 a x-52 a^2 x^2+104 a^3 x^3\right )}{105 x^3 \sqrt {1-a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6684, 6678, 516, 87, 55, 55, 48}

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.

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

\(\Big \downarrow \) 6684

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{-\text {arctanh}(a x)} \sqrt {1-a x}}{x^{9/2}}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 6678

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(1-a x)^{3/2}}{x^{9/2} \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 516

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {1-a x}{x^{9/2} \sqrt {a x+1}}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 48

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

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 516

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

rule 6678

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

rule 6684

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p)   Int[u*(1 + c*(x/d))^p*(E^(n*Ar 
cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 
 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.36

method	result	size
orering	\(-\frac {2 \sqrt {c -\frac {c}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right )}{105 x^{3} \left (a x -1\right )}\)	\(60\)
gosper	\(-\frac {2 \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{105 x^{3} \left (a x -1\right )}\)	\(62\)
default	\(-\frac {2 \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{105 x^{3} \left (a x -1\right )}\)	\(62\)
risch	\(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (104 a^{4} x^{4}+52 a^{3} x^{3}-13 a^{2} x^{2}+24 a x -15\right )}{105 \sqrt {-a^{2} x^{2}+1}\, x^{3} \sqrt {-\left (a x +1\right ) a c x}}\)	\(98\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2 \, {\left (104 \, a^{3} x^{3} - 52 \, a^{2} x^{2} + 39 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{4}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{4}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {152\,\sqrt {1-a^2\,x^2}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )}-\frac {26\,\sqrt {1-a^2\,x^2}\,\left (8\,a^2\,x^2+4\,a\,x+7\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3} \] Input:

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

Output:

- (152*(1 - a^2*x^2)^(1/2)*((c*(a*x - 1))/(a*x))^(1/2))/(105*x^3*(a*x - 1) 
) - (26*(1 - a^2*x^2)^(1/2)*(4*a*x + 8*a^2*x^2 + 7)*((c*(a*x - 1))/(a*x))^ 
(1/2))/(105*x^3)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c}\, i \left (-104 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+52 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-39 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +15 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+104 a^{4} x^{4}\right )}{105 a \,x^{4}} \] Input:

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

Output:

(2*sqrt(c)*i*( - 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 + 1 
5*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 104*a**4*x**4))/(105*a*x**4)