Integrand size = 14, antiderivative size = 271 \[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac {1}{2} a \arctan \left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt {2}}-\frac {1}{2} a \text {arctanh}\left (\frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac {a \log \left (1-\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt {2}} \]
-(-a*x+1)^(7/8)*(a*x+1)^(1/8)/x-1/2*a*arctan((a*x+1)^(1/8)/(-a*x+1)^(1/8)) -1/2*a*arctanh((a*x+1)^(1/8)/(-a*x+1)^(1/8))+1/4*a*arctan(1-(a*x+1)^(1/8)* 2^(1/2)/(-a*x+1)^(1/8))*2^(1/2)-1/4*a*arctan(1+(a*x+1)^(1/8)*2^(1/2)/(-a*x +1)^(1/8))*2^(1/2)+1/8*a*ln(1+(a*x+1)^(1/4)/(-a*x+1)^(1/4)-(a*x+1)^(1/8)*2 ^(1/2)/(-a*x+1)^(1/8))*2^(1/2)-1/8*a*ln(1+(a*x+1)^(1/4)/(-a*x+1)^(1/4)+(a* x+1)^(1/8)*2^(1/2)/(-a*x+1)^(1/8))*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.21 \[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=-\frac {(1-a x)^{7/8} \left (7+7 a x+2 a x \operatorname {Hypergeometric2F1}\left (\frac {7}{8},1,\frac {15}{8},\frac {1-a x}{1+a x}\right )\right )}{7 x (1+a x)^{7/8}} \]
-1/7*((1 - a*x)^(7/8)*(7 + 7*a*x + 2*a*x*Hypergeometric2F1[7/8, 1, 15/8, ( 1 - a*x)/(1 + a*x)]))/(x*(1 + a*x)^(7/8))
Time = 0.45 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 105, 104, 758, 755, 756, 216, 219, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {\sqrt [8]{a x+1}}{x^2 \sqrt [8]{1-a x}}dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {1}{4} a \int \frac {1}{x \sqrt [8]{1-a x} (a x+1)^{7/8}}dx-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle 2 a \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle 2 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-1}d\left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}{\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1}d\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 a \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {\sqrt {2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}\) |
-(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/x) + 2*a*((-1/2*ArcTan[(1 + a*x)^(1/8) /(1 - a*x)^(1/8)] - ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)]/2)/2 + ((ArcT an[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)]/Sqrt[2] - ArcTan[1 + (Sq rt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/(2*Sqrt[2 ]) - Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/( 1 - a*x)^(1/4)]/(2*Sqrt[2]))/2)/2)
3.2.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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])
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])
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])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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]]))
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]
Int[((a_) + (b_.)*(x_)^(n_))^(-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^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
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]
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]
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]
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]
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]
\[\int \frac {{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {1}{4}}}{x^{2}}d x\]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=-\frac {2 \, a x \arctan \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + a x \log \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + 1\right ) - a x \log \left (\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} - 1\right ) + \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + \left (-a^{4}\right )^{\frac {1}{4}}\right ) + i \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} + i \, \left (-a^{4}\right )^{\frac {1}{4}}\right ) - i \, \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} - i \, \left (-a^{4}\right )^{\frac {1}{4}}\right ) - \left (-a^{4}\right )^{\frac {1}{4}} x \log \left (a \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}} - \left (-a^{4}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (a x - 1\right )} \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}}{4 \, x} \]
-1/4*(2*a*x*arctan((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + a*x*log((-sqrt (-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 1) - a*x*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1) + (-a^4)^(1/4)*x*log(a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/ 4) + (-a^4)^(1/4)) + I*(-a^4)^(1/4)*x*log(a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1) )^(1/4) + I*(-a^4)^(1/4)) - I*(-a^4)^(1/4)*x*log(a*(-sqrt(-a^2*x^2 + 1)/(a *x - 1))^(1/4) - I*(-a^4)^(1/4)) - (-a^4)^(1/4)*x*log(a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - (-a^4)^(1/4)) - 4*(a*x - 1)*(-sqrt(-a^2*x^2 + 1)/(a* x - 1))^(1/4))/x
\[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=\int \frac {\sqrt [4]{\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{x^{2}}\, dx \]
\[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x^{2}} \,d x } \]
\[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{\frac {1}{4} \text {arctanh}(a x)}}{x^2} \, dx=\int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4}}{x^2} \,d x \]