\(\int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx\) [530]

Optimal result

Integrand size = 25, antiderivative size = 124 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=-\frac {7 c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 (-7+2 a x)}{-4+4 a x}-\frac {7 \sqrt {c} \log (1-a x)}{8 a^2}+\frac {7 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{8 a^2} \] Input:

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

Output:

(Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-7 + 2*a*x))/(-4 + 4*a*x) - 
(7*Sqrt[c]*Log[1 - a*x])/(8*a^2) + (7*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1 
/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(8*a^2)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6733, 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.

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

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 580

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

\(\Big \downarrow \) 579

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

\(\Big \downarrow \) 573

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((-1/2*(c^2*Sqrt[1 - 1/(a^2*x^2)]*x^2)/Sqrt[c - c/(a*x)] - (7*c*(-((c*Sqr 
t[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)]) + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[ 
1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a))/(4*a))/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 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 = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-14 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{8 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {x \left (a x +1\right )}}\)	\(116\)
risch	\(\frac {\left (2 a x -7\right ) \left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{4 a \left (a x -1\right )}+\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{8 a \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(152\)

Input:

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

Output:

1/8*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)*(4*a^( 
3/2)*x*(x*(a*x+1))^(1/2)-14*(x*(a*x+1))^(1/2)*a^(1/2)+7*ln(1/2*(2*(x*(a*x+ 
1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/(a*x-1)/(x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.59 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (2 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} x - a^{2}\right )}}, -\frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (2 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} x - a^{2}\right )}}\right ] \] Input:

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

Output:

[1/16*(7*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3* 
a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - 
 c)/(a*x - 1)) + 4*(2*a^3*x^3 - 5*a^2*x^2 - 7*a*x)*sqrt((a*x - 1)/(a*x + 1 
))*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2), -1/8*(7*(a*x - 1)*sqrt(-c)*arct 
an(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/( 
a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(2*a^3*x^3 - 5*a^2*x^2 - 7*a*x)*sqrt( 
(a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.40 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -7 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+7 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{4 a^{2}} \] Input:

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

Output:

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