\(\int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx\) [89]

Optimal result

Integrand size = 10, antiderivative size = 130 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {10 \sqrt [4]{1+\frac {1}{a x}}}{a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{a}+\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{a} \] Output:

-10*(1+1/a/x)^(1/4)/a/(1-1/a/x)^(1/4)+(1+1/a/x)^(5/4)*x/(1-1/a/x)^(1/4)+5* 
arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a+5*arctanh((1+1/a/x)^(1/4)/(1-1/a 
/x)^(1/4))/a
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.52 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {-\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (-5+4 e^{2 \coth ^{-1}(a x)}\right )}{-1+e^{2 \coth ^{-1}(a x)}}+5 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+5 \text {arctanh}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{a} \] Input:

Integrate[E^((5*ArcCoth[a*x])/2),x]
 

Output:

((-2*E^(ArcCoth[a*x]/2)*(-5 + 4*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a* 
x])) + 5*ArcTan[E^(ArcCoth[a*x]/2)] + 5*ArcTanh[E^(ArcCoth[a*x]/2)])/a
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6720, 105, 105, 104, 756, 216, 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.

\(\displaystyle \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6720

\(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^{5/4} x^2}{\left (1-\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \int \frac {\sqrt [4]{1+\frac {1}{a x}} x}{\left (1-\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{2 a}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \left (\int \frac {x}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {4 \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \left (4 \int \frac {1}{\frac {1}{x^4}-1}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {4 \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \left (4 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4 \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \left (4 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4 \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \left (4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4 \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}\)

Input:

Int[E^((5*ArcCoth[a*x])/2),x]
 

Output:

((1 + 1/(a*x))^(5/4)*x)/(1 - 1/(a*x))^(1/4) - (5*((4*(1 + 1/(a*x))^(1/4))/ 
(1 - 1/(a*x))^(1/4) + 4*(-1/2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/ 
4)] - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/2)))/(2*a)
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 6720

Int[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/( 
x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n]
 

Maple [F]

\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]

Input:

int(1/((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int(1/((a*x-1)/(a*x+1))^(5/4),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {10 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 5 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (a^{2} x^{2} - 8 \, a x - 9\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{2 \, {\left (a^{2} x - a\right )}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="fricas")
 

Output:

-1/2*(10*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 5*(a*x - 1)*log(( 
(a*x - 1)/(a*x + 1))^(1/4) + 1) + 5*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1 
/4) - 1) - 2*(a^2*x^2 - 8*a*x - 9)*((a*x - 1)/(a*x + 1))^(3/4))/(a^2*x - a 
)
 

Sympy [F]

\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=\int \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \] Input:

integrate(1/((a*x-1)/(a*x+1))**(5/4),x)
 

Output:

Integral(((a*x - 1)/(a*x + 1))**(-5/4), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.01 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {1}{2} \, a {\left (\frac {4 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {10 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{2}} - \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} + \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{2}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="maxima")
 

Output:

-1/2*a*(4*(5*(a*x - 1)/(a*x + 1) - 4)/(a^2*((a*x - 1)/(a*x + 1))^(5/4) - a 
^2*((a*x - 1)/(a*x + 1))^(1/4)) + 10*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a 
^2 - 5*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^2 + 5*log(((a*x - 1)/(a*x + 
1))^(1/4) - 1)/a^2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {1}{2} \, a {\left (\frac {10 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{2}} - \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} + \frac {5 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{2}} + \frac {4 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2} {\left (\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="giac")
 

Output:

-1/2*a*(10*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^2 - 5*log(((a*x - 1)/(a*x 
 + 1))^(1/4) + 1)/a^2 + 5*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^2 + 
4*(5*(a*x - 1)/(a*x + 1) - 4)/(a^2*((a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4)/ 
(a*x + 1) - ((a*x - 1)/(a*x + 1))^(1/4))))
 

Mupad [B] (verification not implemented)

Time = 23.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {\frac {10\,\left (a\,x-1\right )}{a\,x+1}-8}{a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}-\frac {5\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{a}+\frac {5\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{a} \] Input:

int(1/((a*x - 1)/(a*x + 1))^(5/4),x)
 

Output:

((10*(a*x - 1))/(a*x + 1) - 8)/(a*((a*x - 1)/(a*x + 1))^(1/4) - a*((a*x - 
1)/(a*x + 1))^(5/4)) - (5*atan(((a*x - 1)/(a*x + 1))^(1/4)))/a + (5*atanh( 
((a*x - 1)/(a*x + 1))^(1/4)))/a
 

Reduce [F]

\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{4}}}{\left (a x -1\right )^{\frac {1}{4}} a x -\left (a x -1\right )^{\frac {1}{4}}}d x +\left (\int \frac {\left (a x +1\right )^{\frac {1}{4}} x}{\left (a x -1\right )^{\frac {1}{4}} a x -\left (a x -1\right )^{\frac {1}{4}}}d x \right ) a \] Input:

int(1/((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int((a*x + 1)**(1/4)/((a*x - 1)**(1/4)*a*x - (a*x - 1)**(1/4)),x) + int((( 
a*x + 1)**(1/4)*x)/((a*x - 1)**(1/4)*a*x - (a*x - 1)**(1/4)),x)*a