\(\int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) [458]

Optimal result

Integrand size = 22, antiderivative size = 134 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (\sqrt {1+\frac {1}{a x}} x+\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a}\right )}{\sqrt {c-\frac {c}{a x}}} \] Input:

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

Output:

(Sqrt[1 - 1/(a*x)]*(Sqrt[1 + 1/(a*x)]*x + (3*ArcTanh[Sqrt[1 + 1/(a*x)]])/a 
 - (2*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/a))/Sqrt[c - c/(a*x)]
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6731, 585, 27, 110, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \sqrt {1-\frac {1}{a x}} \left (\frac {8 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+6 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{2 a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{a}\right )}{\sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \sqrt {1-\frac {1}{a x}} \left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-6 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{2 a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{a}\right )}{\sqrt {c-\frac {c}{a x}}}\)

Input:

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

Output:

-((a*Sqrt[1 - 1/(a*x)]*(-((Sqrt[1 + 1/(a*x)]*x)/a) + (-6*ArcTanh[Sqrt[1 + 
1/(a*x)]] + 4*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(2*a^2)))/Sqrt[c 
 - c/(a*x)])
 

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 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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 585

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

rule 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^n   Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ 
x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] 
&& IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {3}{2}} c \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\)	\(151\)
risch	\(\frac {a x -1}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a \sqrt {a^{2} c}}-\frac {\sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a^{2} \sqrt {c}}\right ) \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\)	\(223\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (111) = 222\).

Time = 0.15 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.86 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} + \frac {2 \, \sqrt {2} {\left (a c x - c\right )} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac {4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt {c}}}{4 \, {\left (a^{2} c x - a c\right )}}, \frac {2 \, \sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - 3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \] Input:

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

Output:

[1/4*(3*(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*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - 
c)/(a*x)) + 2*sqrt(2)*(a*c*x - c)*log(-(17*a^3*x^3 - 3*a^2*x^2 - 13*a*x - 
4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a* 
c*x - c)/(a*x))/sqrt(c) - 1)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1))/sqrt(c))/( 
a^2*c*x - a*c), 1/2*(2*sqrt(2)*(a*c*x - c)*sqrt(-1/c)*arctan(2*sqrt(2)*(a^ 
2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x))/ 
(3*a^2*x^2 - 2*a*x - 1)) - 3*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*s 
qrt(-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*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/ 
(a*x)))/(a^2*c*x - a*c)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+\sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+\sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+3 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{a c} \] Input:

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

Output:

(sqrt(c)*(sqrt(x)*sqrt(a)*sqrt(a*x + 1) + sqrt(2)*log(sqrt(a*x + 1) + sqrt 
(x)*sqrt(a) - sqrt(2) - 1) - sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - 
 sqrt(2) + 1) - sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1) 
 + sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1) + 3*log(sqrt 
(a*x + 1) + sqrt(x)*sqrt(a))))/(a*c)