Integrand size = 14, antiderivative size = 253 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {611 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{1920 a^4}-\frac {269 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac {9 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5+\frac {31 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {31 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \]
611/1920*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x/a^4-269/960*(1-1/a/x)^(1/4)*(1+ 1/a/x)^(3/4)*x^2/a^3+11/48*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^3/a^2-9/40*(1 -1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^4/a+1/5*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^ 5+31/128*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5-31/128*arctanh((1+1/a /x)^(1/4)/(1-1/a/x)^(1/4))/a^5
Time = 5.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.68 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {24576 e^{\frac {19}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}-\frac {62976 e^{\frac {15}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {64640 e^{\frac {11}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}-\frac {34000 e^{\frac {7}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {9620 e^{\frac {3}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}-930 \arctan \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )+465 \log \left (1-e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )-465 \log \left (1+e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )}{3840 a^5} \]
((24576*E^((19*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 - (62976*E^(( 15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (64640*E^((11*ArcCoth[a *x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 - (34000*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (9620*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth [a*x])) - 930*ArcTan[E^(-1/2*ArcCoth[a*x])] + 465*Log[1 - E^(-1/2*ArcCoth[ a*x])] - 465*Log[1 + E^(-1/2*ArcCoth[a*x])])/(3840*a^5)
Time = 0.41 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.08, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {6721, 110, 27, 168, 27, 168, 27, 168, 27, 168, 27, 104, 25, 827, 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.
| \(\displaystyle \int x^4 e^{-\frac {1}{2} \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [4]{1-\frac {1}{a x}} x^6}{\sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {1}{5} \int -\frac {\left (9 a-\frac {8}{x}\right ) x^5}{2 a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (9 a-\frac {8}{x}\right ) x^5}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {1}{4} \int \frac {\left (55 a-\frac {54}{x}\right ) x^4}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (55 a-\frac {54}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {\left (269 a-\frac {220}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (269 a-\frac {220}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {1}{2} \int \frac {\left (611 a-\frac {538}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {\left (611 a-\frac {538}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-\int \frac {465 x}{2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-\frac {465}{2} \int \frac {x}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-930 \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {930 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-930 \left (\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-930 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {-930 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )-611 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {269}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {55}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{8 a}-\frac {9}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}+\frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\) |
((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5)/5 + ((-9*a*(1 - 1/(a*x))^(1/ 4)*(1 + 1/(a*x))^(3/4)*x^4)/4 - ((-55*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^ (3/4)*x^3)/3 - ((-269*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/2 - ( -611*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x - 930*(ArcTan[(1 + 1/(a*x ))^(1/4)/(1 - 1/(a*x))^(1/4)]/2 - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x) )^(1/4)]/2))/(4*a))/(6*a))/(8*a))/(10*a^2)
3.1.86.3.1 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[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])
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]
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[(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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x /a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n] && IntegerQ[m]
\[\int x^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}d x\]
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.47 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {2 \, {\left (384 \, a^{5} x^{5} - 48 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 98 \, a^{2} x^{2} + 73 \, a x + 611\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{3840 \, a^{5}} \]
1/3840*(2*(384*a^5*x^5 - 48*a^4*x^4 + 8*a^3*x^3 - 98*a^2*x^2 + 73*a*x + 61 1)*((a*x - 1)/(a*x + 1))^(1/4) - 930*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 465*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 465*log(((a*x - 1)/(a*x + 1))^ (1/4) - 1))/a^5
\[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\int x^{4} \sqrt [4]{\frac {a x - 1}{a x + 1}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (2405 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 1120 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 5090 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 696 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 465 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac {930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \]
-1/3840*a*(4*(2405*((a*x - 1)/(a*x + 1))^(17/4) - 1120*((a*x - 1)/(a*x + 1 ))^(13/4) + 5090*((a*x - 1)/(a*x + 1))^(9/4) - 696*((a*x - 1)/(a*x + 1))^( 5/4) + 465*((a*x - 1)/(a*x + 1))^(1/4))/(5*(a*x - 1)*a^6/(a*x + 1) - 10*(a *x - 1)^2*a^6/(a*x + 1)^2 + 10*(a*x - 1)^3*a^6/(a*x + 1)^3 - 5*(a*x - 1)^4 *a^6/(a*x + 1)^4 + (a*x - 1)^5*a^6/(a*x + 1)^5 - a^6) + 930*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 465*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 - 465*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6)
Time = 0.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {465 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} - \frac {4 \, {\left (\frac {696 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {5090 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {1120 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {2405 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{4}} - 465 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \]
-1/3840*a*(930*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 465*log(((a*x - 1 )/(a*x + 1))^(1/4) + 1)/a^6 - 465*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1) )/a^6 - 4*(696*(a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 5090*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 + 1120*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^3 - 2405*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^ (1/4)/(a*x + 1)^4 - 465*((a*x - 1)/(a*x + 1))^(1/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))
Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {31\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{64}-\frac {29\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{40}+\frac {509\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{96}-\frac {7\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{6}+\frac {481\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}}{192}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}-\frac {31\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {31\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5} \]
((31*((a*x - 1)/(a*x + 1))^(1/4))/64 - (29*((a*x - 1)/(a*x + 1))^(5/4))/40 + (509*((a*x - 1)/(a*x + 1))^(9/4))/96 - (7*((a*x - 1)/(a*x + 1))^(13/4)) /6 + (481*((a*x - 1)/(a*x + 1))^(17/4))/192)/(a^5 + (10*a^5*(a*x - 1)^2)/( a*x + 1)^2 - (10*a^5*(a*x - 1)^3)/(a*x + 1)^3 + (5*a^5*(a*x - 1)^4)/(a*x + 1)^4 - (a^5*(a*x - 1)^5)/(a*x + 1)^5 - (5*a^5*(a*x - 1))/(a*x + 1)) - (31 *atan(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5) - (31*atanh(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5)