\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx\) [67]

Optimal result

Integrand size = 14, antiderivative size = 168 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x^4}-\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x^3}-\frac {29 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{96 x^2}-\frac {83 a^3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{192 x}-\frac {11}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {11}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \] Output:

-1/4*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/x^4-7/24*a*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/ 
x^3-29/96*a^2*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/x^2-83/192*a^3*(-a*x+1)^(3/4)*( 
a*x+1)^(1/4)/x-11/64*a^4*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))-11/64*a^4*ar 
ctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {(1-a x)^{3/4} \left (48+104 a x+114 a^2 x^2+141 a^3 x^3+83 a^4 x^4+22 a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )\right )}{192 x^4 (1+a x)^{3/4}} \] Input:

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

Output:

-1/192*((1 - a*x)^(3/4)*(48 + 104*a*x + 114*a^2*x^2 + 141*a^3*x^3 + 83*a^4 
*x^4 + 22*a^4*x^4*Hypergeometric2F1[3/4, 1, 7/4, (1 - a*x)/(1 + a*x)]))/(x 
^4*(1 + a*x)^(3/4))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6676, 110, 27, 168, 27, 168, 27, 168, 27, 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.

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

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {\sqrt [4]{a x+1}}{x^5 \sqrt [4]{1-a x}}dx\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{4} \int \frac {a (6 a x+7)}{2 x^4 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \int \frac {6 a x+7}{x^4 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (-\frac {1}{3} \int -\frac {a (28 a x+29)}{2 x^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \int \frac {28 a x+29}{x^3 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {a (58 a x+83)}{2 x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \int \frac {58 a x+83}{x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (-\int -\frac {33 a}{2 x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {33}{2} a \int \frac {1}{x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (66 a \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (66 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (66 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (66 a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {83 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {29 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{4 x^4}\)

Input:

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

Output:

-1/4*((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x^4 + (a*((-7*(1 - a*x)^(3/4)*(1 + 
a*x)^(1/4))/(3*x^3) + (a*((-29*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(2*x^2) + 
(a*((-83*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x + 66*a*(-1/2*ArcTan[(1 + a*x)^ 
(1/4)/(1 - a*x)^(1/4)] - ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)))/4)) 
/6))/8
 

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 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 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 168

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] && ILtQ[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 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{5}}d x\]

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x)
 

Output:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {66 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 33 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (83 \, a^{4} x^{4} - 25 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 8 \, a x - 48\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/384*(66*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 33*a^4*x^ 
4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 33*a^4*x^4*log(sqrt(-sqrt 
(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(83*a^4*x^4 - 25*a^3*x^3 - 2*a^2*x^2 - 
8*a*x - 48)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4
 

Sympy [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{x^{5}}\, dx \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**5,x)
 

Output:

Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**5, x)
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{x^{5}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x, algorithm="maxima")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/x^5, x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{x^{5}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x, algorithm="giac")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^5} \,d x \] Input:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/x^5,x)
 

Output:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/x^5, x)
 

Reduce [F]

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

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x)
 

Output:

int(sqrt(a*x + 1)/(( - a**2*x**2 + 1)**(1/4)*x**5),x)