\(\int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [539]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\sqrt {c-\frac {c}{a x}} x-\frac {5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Output:

(c-c/a/x)^(1/2)*x-5*c^(1/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a+4*2^(1/2)*c 
^(1/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\sqrt {c-\frac {c}{a x}} x-\frac {5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Input:

Integrate[Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]),x]
 

Output:

Sqrt[c - c/(a*x)]*x - (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + ( 
4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6717, 6683, 1035, 281, 899, 109, 27, 174, 73, 221}

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.

\(\displaystyle \int \sqrt {c-\frac {c}{a x}} e^{-2 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

\(\displaystyle -\int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}}}{a+\frac {1}{x}}dx\)

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle -\frac {a \left (-\frac {c^2 \left (5 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-8 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \left (-\frac {c^2 \left (\frac {16 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {10 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \left (-\frac {c^2 \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\)

Input:

Int[Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]),x]
 

Output:

-((a*(-((c*Sqrt[c - c/(a*x)]*x)/a) - (c^2*((-10*ArcTanh[Sqrt[c - c/(a*x)]/ 
Sqrt[c]])/Sqrt[c] + (8*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c]) 
])/Sqrt[c]))/(2*a^2)))/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 109

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

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

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

rule 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

rule 1035

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) 
 + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c 
 + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ 
mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

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

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(75)=150\).

Time = 0.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.83

method	result	size
risch	\(x \sqrt {\frac {c \left (a x -1\right )}{a x}}+\frac {\left (-\frac {5 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a \sqrt {c}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\)	\(168\)
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-4 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}-6 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{2 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}}\)	\(189\)

Input:

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

Output:

x*(c*(a*x-1)/a/x)^(1/2)+(-5/2*ln((-1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x 
^2-a*c*x)^(1/2))/(a^2*c)^(1/2)-2/a*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2 
*2^(1/2)*c^(1/2)*((x+1/a)^2*a^2*c-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))*(c*( 
a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.57 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} + 4 \, \sqrt {2} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 5 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, \frac {a x \sqrt {\frac {a c x - c}{a x}} - 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 5 \, \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right )}{a}\right ] \] Input:

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

Output:

[1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) + 4*sqrt(2)*sqrt(c)*log(-(2*sqrt(2)*a* 
sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) + 5*sqrt(c)*lo 
g(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, (a*x*sqrt((a*c 
*x - c)/(a*x)) - 4*sqrt(2)*sqrt(-c)*arctan(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c* 
x - c)/(a*x))/(a*c*x - c)) + 5*sqrt(-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - 
c)/(a*x))/(a*c*x - c)))/a]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )+2 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )-2 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )-5 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )\right )}{a} \] Input:

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

Output:

(sqrt(c)*(sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2)*log(sqrt(a*x - 1) + sq 
rt(x)*sqrt(a) - sqrt(2)*i + i) + 2*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqr 
t(a) + sqrt(2)*i - i) - 2*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2* 
sqrt(2) + 2*a*x + 2) - 5*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))))/a