3.2.8 \(\int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx\) [108]

3.2.8.1 Optimal result

Integrand size = 14, antiderivative size = 139 \[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=-\frac {(1-a x)^{3/4} \sqrt [4]{1+a x}}{3 x^3}+\frac {7 a (1-a x)^{3/4} \sqrt [4]{1+a x}}{12 x^2}-\frac {23 a^2 (1-a x)^{3/4} \sqrt [4]{1+a x}}{24 x}+\frac {17}{8} a^3 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {17}{8} a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]

output

-1/3*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/x^3+7/12*a*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/ 
x^2-23/24*a^2*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/x+17/8*a^3*arctan((a*x+1)^(1/4) 
/(-a*x+1)^(1/4))+17/8*a^3*arctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))
 

3.2.8.2 Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\frac {(1-a x)^{3/4} \left (-8+6 a x-9 a^2 x^2-23 a^3 x^3+34 a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )\right )}{24 x^3 (1+a x)^{3/4}} \]

input

Integrate[1/(E^((3*ArcTanh[a*x])/2)*x^4),x]
 

output

((1 - a*x)^(3/4)*(-8 + 6*a*x - 9*a^2*x^2 - 23*a^3*x^3 + 34*a^3*x^3*Hyperge 
ometric2F1[3/4, 1, 7/4, (1 - a*x)/(1 + a*x)]))/(24*x^3*(1 + a*x)^(3/4))
 

3.2.8.3 Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6676, 110, 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.

\(\displaystyle \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {(1-a x)^{3/4}}{x^4 (a x+1)^{3/4}}dx\)

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{2} \int \frac {a (23-14 a x)}{2 x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \int \frac {23-14 a x}{x^2 \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-\int \frac {51 a}{2 x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-\frac {51}{2} a \int \frac {1}{x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-102 a \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-102 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-102 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {1}{6} a \left (-\frac {1}{4} a \left (-102 a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {23 (1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {7 (1-a x)^{3/4} \sqrt [4]{a x+1}}{2 x^2}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{3 x^3}\)

input

Int[1/(E^((3*ArcTanh[a*x])/2)*x^4),x]
 

output

-1/3*((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x^3 - (a*((-7*(1 - a*x)^(3/4)*(1 + 
a*x)^(1/4))/(2*x^2) - (a*((-23*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x - 102*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
 

3.2.8.3.1 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]
 

3.2.8.4 Maple [F]

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

input

int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x)
 

output

int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x)
 

3.2.8.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\frac {102 \, a^{3} x^{3} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, {\left (23 \, a^{3} x^{3} - 37 \, a^{2} x^{2} + 22 \, a x - 8\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}} \]

input

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm="fricas")
 

output

1/48*(102*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 51*a^3*x^3 
*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 51*a^3*x^3*log(sqrt(-sqrt( 
-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(23*a^3*x^3 - 37*a^2*x^2 + 22*a*x - 8)*s 
qrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3
 

3.2.8.6 Sympy [F]

\[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**4,x)
 

output

Integral(1/(x**4*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)
 

3.2.8.7 Maxima [F]

\[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\int { \frac {1}{x^{4} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}} \,d x } \]

input

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm="maxima")
 

output

integrate(1/(x^4*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(3/2)), x)
 

3.2.8.8 Giac [F]

\[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\int { \frac {1}{x^{4} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}} \,d x } \]

input

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm="giac")
 

output

integrate(1/(x^4*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(3/2)), x)
 

3.2.8.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {3}{2} \text {arctanh}(a x)}}{x^4} \, dx=\int \frac {1}{x^4\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}} \,d x \]

input

int(1/(x^4*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(3/2)),x)
 

output

int(1/(x^4*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(3/2)), x)