\(\int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [769]

Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \] Output:

1/3*(-4*a-4/x)/a^2/c/(1-1/a^2/x^2)^(3/2)-1/3*(9*a+11/x)/a^2/c/(1-1/a^2/x^2 
)^(1/2)+(1-1/a^2/x^2)^(1/2)*x/c+3*arctanh((1-1/a^2/x^2)^(1/2))/a/c
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (14-19 a x+3 a^2 x^2\right )}{(-1+a x)^2}+\frac {9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \] Input:

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]
 

Output:

((Sqrt[1 - 1/(a^2*x^2)]*x*(14 - 19*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + (9*Log 
[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(3*c)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6748, 110, 27, 169, 25, 27, 169, 27, 103, 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.

\(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle -\frac {\frac {\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {1}{3} a \int -\frac {\left (9 a+\frac {5}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {1}{3} a \int \frac {\left (9 a+\frac {5}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {1}{3} \int \frac {\left (9 a+\frac {5}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {14 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}-a \int -\frac {9 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {1}{3} \left (9 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {14 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}\right )+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 103

\(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {14 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}-9 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )\right )+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {14 a \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}-9 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+\frac {5 a \sqrt {\frac {1}{a x}+1}}{3 \left (1-\frac {1}{a x}\right )^{3/2}}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\left (1-\frac {1}{a x}\right )^{3/2}}}{c}\)

Input:

Int[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]
 

Output:

-((-((Sqrt[1 + 1/(a*x)]*x)/(1 - 1/(a*x))^(3/2)) + ((5*a*Sqrt[1 + 1/(a*x)]) 
/(3*(1 - 1/(a*x))^(3/2)) + ((14*a*Sqrt[1 + 1/(a*x)])/Sqrt[1 - 1/(a*x)] - 9 
*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/3)/a^2)/c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 103

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

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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 6748

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.76

method	result	size
risch	\(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {13 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(185\)
default	\(-\frac {-9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+9 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{3 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(346\)

Input:

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

Output:

1/a*(a*x-1)/c/((a*x-1)/(a*x+1))^(1/2)+(3/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2 
-1)^(1/2))/(a^2)^(1/2)-13/3/a^4/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)- 
2/3/a^5/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2))/c*a^2/((a*x-1)/(a*x+1 
))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.22 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \] Input:

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

Output:

1/3*(9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a^2*x 
^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 16*a^2*x 
^2 - 5*a*x + 14)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.27 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {1}{3} \, a {\left (\frac {\frac {11 \, {\left (a x - 1\right )}}{a x + 1} - \frac {18 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \] Input:

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

Output:

1/3*a*((11*(a*x - 1)/(a*x + 1) - 18*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c*(( 
a*x - 1)/(a*x + 1))^(5/2) - a^2*c*((a*x - 1)/(a*x + 1))^(3/2)) + 9*log(sqr 
t((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(sqrt((a*x - 1)/(a*x + 1)) - 1) 
/(a^2*c))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 13.49 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {11\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \] Input:

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

Output:

(6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c) - ((11*(a*x - 1))/(3*(a*x + 1 
)) - (6*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(a*c*((a*x - 1)/(a*x + 1))^(3/2) - 
 a*c*((a*x - 1)/(a*x + 1))^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {18 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -18 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+4 \sqrt {a x -1}\, a x -4 \sqrt {a x -1}+3 \sqrt {a x +1}\, a^{2} x^{2}-19 \sqrt {a x +1}\, a x +14 \sqrt {a x +1}}{3 \sqrt {a x -1}\, a c \left (a x -1\right )} \] Input:

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

Output:

(18*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 18*sq 
rt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 4*sqrt(a*x - 1) 
*a*x - 4*sqrt(a*x - 1) + 3*sqrt(a*x + 1)*a**2*x**2 - 19*sqrt(a*x + 1)*a*x 
+ 14*sqrt(a*x + 1))/(3*sqrt(a*x - 1)*a*c*(a*x - 1))