\(\int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) [436]

Optimal result

Integrand size = 23, antiderivative size = 112 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=-\frac {c \sqrt {1-a x} \left ((2-7 a x) \sqrt {1+a x}+7 a^2 x^2 \text {arctanh}\left (\sqrt {1+a x}\right )\right )}{4 x^2 \sqrt {c-a c x}} \] Input:

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

Output:

-1/4*(c*Sqrt[1 - a*x]*((2 - 7*a*x)*Sqrt[1 + a*x] + 7*a^2*x^2*ArcTanh[Sqrt[ 
1 + a*x]]))/(x^2*Sqrt[c - a*c*x])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6678, 580, 579, 573, 219}

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.

\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx\)

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 580

\(\displaystyle \frac {-\frac {7}{4} a c \int \frac {\sqrt {c-a c x}}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\)

\(\Big \downarrow \) 579

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

\(\Big \downarrow \) 573

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {7}{4} a c \left (a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\)

Input:

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

Output:

(-1/2*(c^2*Sqrt[1 - a^2*x^2])/(x^2*Sqrt[c - a*c*x]) - (7*a*c*(-((c*Sqrt[1 
- a^2*x^2])/(x*Sqrt[c - a*c*x])) + a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2 
*x^2])/Sqrt[c - a*c*x]]))/4)/c
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 573

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

rule 579

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

rule 580

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

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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 c \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{2} x^{2}-7 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}+2 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{2}}\)	\(101\)
risch	\(-\frac {\left (7 a^{2} x^{2}+5 a x -2\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{4 x^{2} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {7 a^{2} \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\)	\(150\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\left [\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{8 \, {\left (a x^{3} - x^{2}\right )}}, -\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{4 \, {\left (a x^{3} - x^{2}\right )}}\right ] \] Input:

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

Output:

[1/8*(7*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2* 
x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*sqrt(-a^2*x^2 + 
1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2), -1/4*(7*(a^3*x^3 - a^2*x^2 
)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c) 
) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2)]
 

Sympy [F]

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

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

Output:

Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)), x 
)
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {1}{4} \, a^{2} c {\left (\frac {7 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {7 \, {\left (a c x + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {a c x + c} c}{a^{2} c^{3} x^{2}}\right )} {\left | c \right |} - \frac {7 \, a^{2} c {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + 5 \, \sqrt {2} a^{2} \sqrt {-c} \sqrt {c} {\left | c \right |}}{4 \, \sqrt {-c} c} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {\sqrt {c}\, \left (14 \sqrt {a x +1}\, a x -4 \sqrt {a x +1}+7 \,\mathrm {log}\left (\sqrt {a x +1}-1\right ) a^{2} x^{2}-7 \,\mathrm {log}\left (\sqrt {a x +1}+1\right ) a^{2} x^{2}\right )}{8 x^{2}} \] Input:

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

Output:

(sqrt(c)*(14*sqrt(a*x + 1)*a*x - 4*sqrt(a*x + 1) + 7*log(sqrt(a*x + 1) - 1 
)*a**2*x**2 - 7*log(sqrt(a*x + 1) + 1)*a**2*x**2))/(8*x**2)