\(\int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x} \, dx\) [97]

Optimal result

Integrand size = 14, antiderivative size = 195 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x} \, dx=\frac {8 \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-2 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x} \left (1+\frac {\sqrt {1+a x}}{\sqrt {1-a x}}\right )}\right ) \] Output:

8*(a*x+1)^(1/4)/(-a*x+1)^(1/4)-2*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))+2^(1 
/2)*arctan(1-2^(1/2)*(a*x+1)^(1/4)/(-a*x+1)^(1/4))-2^(1/2)*arctan(1+2^(1/2 
)*(a*x+1)^(1/4)/(-a*x+1)^(1/4))-2*arctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))-2^ 
(1/2)*arctanh(2^(1/2)*(a*x+1)^(1/4)/(-a*x+1)^(1/4)/(1+(a*x+1)^(1/2)/(-a*x+ 
1)^(1/2)))
 

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x} \, dx=\frac {4 \left (3+3 a x+3 \sqrt [4]{2} (1+a x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\frac {1}{2} (1-a x)\right )+(-1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )\right )}{3 \sqrt [4]{1-a x} (1+a x)^{3/4}} \] Input:

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

Output:

(4*(3 + 3*a*x + 3*2^(1/4)*(1 + a*x)^(3/4)*Hypergeometric2F1[-1/4, -1/4, 3/ 
4, (1 - a*x)/2] + (-1 + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (1 - a*x)/(1 + 
 a*x)]))/(3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.83 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {6676, 109, 27, 140, 73, 104, 756, 216, 219, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 6676

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle 4 \int \frac {\sqrt {1-a x}}{(a x+1)^{3/4}}d\sqrt [4]{1-a x}+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 854

\(\displaystyle 4 \int \frac {\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 826

\(\displaystyle 4 \left (\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 1476

\(\displaystyle 4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 1082

\(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 217

\(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 1479

\(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 25

\(\displaystyle 4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )+4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 1103

\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {8 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\)

Input:

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

Output:

(8*(1 + a*x)^(1/4))/(1 - a*x)^(1/4) + 4*(-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*((-(ArcTan[1 
 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[ 
2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - a*x] - 
 (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - 
 a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2)
 

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 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 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 109

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

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[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 826

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

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x} \, dx=-\sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {a x + \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\frac {a x - \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right ) + 8 \, \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 2 \, \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) \] Input:

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

Output:

-sqrt(2)*arctan(sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - sqrt(2) 
*arctan(sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 1/2*sqrt(2)*log 
((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2* 
x^2 + 1) - 1)/(a*x - 1)) + 1/2*sqrt(2)*log((a*x - sqrt(2)*(a*x - 1)*sqrt(- 
sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 8*sqr 
t(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x 
- 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^ 
2*x^2 + 1)/(a*x - 1)) - 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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