Integrand size = 14, antiderivative size = 250 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=-\frac {2467 \sqrt [4]{1+\frac {1}{a x}}}{192 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {521 \sqrt [4]{1+\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {113 \sqrt [4]{1+\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {17 \sqrt [4]{1+\frac {1}{a x}} x^3}{24 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^4}{4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {475 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4} \]
-2467/192*(1+1/a/x)^(1/4)/a^4/(1-1/a/x)^(1/4)+521/192*(1+1/a/x)^(1/4)*x/a^ 3/(1-1/a/x)^(1/4)+113/96*(1+1/a/x)^(1/4)*x^2/a^2/(1-1/a/x)^(1/4)+17/24*(1+ 1/a/x)^(1/4)*x^3/a/(1-1/a/x)^(1/4)+1/4*(1+1/a/x)^(1/4)*x^4/(1-1/a/x)^(1/4) +475/64*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^4+475/64*arctanh((1+1/a/ x)^(1/4)/(1-1/a/x)^(1/4))/a^4
Time = 5.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.64 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {-3072 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {1536 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {5248 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {7376 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {6292 e^{\frac {1}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}+2850 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-1425 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+1425 \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{384 a^4} \]
(-3072*E^(ArcCoth[a*x]/2) + (1536*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a *x]))^4 + (5248*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (7376*E^ (ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (6292*E^(ArcCoth[a*x]/2))/ (-1 + E^(2*ArcCoth[a*x])) + 2850*ArcTan[E^(ArcCoth[a*x]/2)] - 1425*Log[1 - E^(ArcCoth[a*x]/2)] + 1425*Log[1 + E^(ArcCoth[a*x]/2)])/(384*a^4)
Time = 0.38 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6721, 109, 27, 168, 27, 168, 27, 168, 27, 172, 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.
| \(\displaystyle \int x^3 e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^{5/4} x^5}{\left (1-\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{4} \int -\frac {\left (17 a+\frac {16}{x}\right ) x^4}{2 a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\int \frac {\left (17 a+\frac {16}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{8 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {-\frac {1}{3} \int -\frac {\left (113 a+\frac {102}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\int \frac {\left (113 a+\frac {102}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {-\frac {1}{2} \int -\frac {\left (521 a+\frac {452}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\int \frac {\left (521 a+\frac {452}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {-\int -\frac {\left (1425 a+\frac {1042}{x}\right ) x}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\int \frac {\left (1425 a+\frac {1042}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}-2 a \int -\frac {1425 x}{2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {1425 a \int \frac {x}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {5700 a \int \frac {1}{\frac {1}{x^4}-1}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {5700 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {5700 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {5700 a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4934 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {113 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {17 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a^2}\) |
((1 + 1/(a*x))^(1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) - ((-17*a*(1 + 1/(a*x))^ (1/4)*x^3)/(3*(1 - 1/(a*x))^(1/4)) + ((-113*a*(1 + 1/(a*x))^(1/4)*x^2)/(2* (1 - 1/(a*x))^(1/4)) + ((-521*a*(1 + 1/(a*x))^(1/4)*x)/(1 - 1/(a*x))^(1/4) + ((4934*a*(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4) + 5700*a*(-1/2*ArcTan [(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/2))/(2*a))/(4*a))/(6*a))/(8*a^2)
3.1.78.3.1 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[(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])
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && 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[(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[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[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]
\[\int \frac {x^{3}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]
Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.58 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=-\frac {2850 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 1425 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 1425 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 362 \, a^{3} x^{3} + 747 \, a^{2} x^{2} - 1946 \, a x - 2467\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{384 \, {\left (a^{5} x - a^{4}\right )}} \]
-1/384*(2850*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 1425*(a*x - 1 )*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 1425*(a*x - 1)*log(((a*x - 1)/(a* x + 1))^(1/4) - 1) - 2*(48*a^5*x^5 + 184*a^4*x^4 + 362*a^3*x^3 + 747*a^2*x ^2 - 1946*a*x - 2467)*((a*x - 1)/(a*x + 1))^(3/4))/(a^5*x - a^4)
\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\int \frac {x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.95 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {1}{384} \, a {\left (\frac {4 \, {\left (\frac {4645 \, {\left (a x - 1\right )}}{a x + 1} - \frac {7483 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {5415 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {1425 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 768\right )}}{a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 4 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 6 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 4 \, a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} + \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} - \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{5}}\right )} \]
1/384*a*(4*(4645*(a*x - 1)/(a*x + 1) - 7483*(a*x - 1)^2/(a*x + 1)^2 + 5415 *(a*x - 1)^3/(a*x + 1)^3 - 1425*(a*x - 1)^4/(a*x + 1)^4 - 768)/(a^5*((a*x - 1)/(a*x + 1))^(17/4) - 4*a^5*((a*x - 1)/(a*x + 1))^(13/4) + 6*a^5*((a*x - 1)/(a*x + 1))^(9/4) - 4*a^5*((a*x - 1)/(a*x + 1))^(5/4) + a^5*((a*x - 1) /(a*x + 1))^(1/4)) - 2850*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^5 + 1425*l og(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 - 1425*log(((a*x - 1)/(a*x + 1))^( 1/4) - 1)/a^5)
Time = 0.35 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.89 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=-\frac {1}{384} \, a {\left (\frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} - \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} + \frac {1425 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{5}} + \frac {3072}{a^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {4 \, {\left (\frac {2875 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {2343 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {657 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - 1573 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{5} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \]
-1/384*a*(2850*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^5 - 1425*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 + 1425*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^5 + 3072/(a^5*((a*x - 1)/(a*x + 1))^(1/4)) + 4*(2875*(a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 2343*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^ (3/4)/(a*x + 1)^2 + 657*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^ 3 - 1573*((a*x - 1)/(a*x + 1))^(3/4))/(a^5*((a*x - 1)/(a*x + 1) - 1)^4))
Time = 0.14 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.84 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {475\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{64\,a^4}-\frac {475\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{64\,a^4}-\frac {\frac {7483\,{\left (a\,x-1\right )}^2}{96\,{\left (a\,x+1\right )}^2}-\frac {1805\,{\left (a\,x-1\right )}^3}{32\,{\left (a\,x+1\right )}^3}+\frac {475\,{\left (a\,x-1\right )}^4}{32\,{\left (a\,x+1\right )}^4}-\frac {4645\,\left (a\,x-1\right )}{96\,\left (a\,x+1\right )}+8}{a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-4\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+6\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-4\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}+a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}} \]
(475*atanh(((a*x - 1)/(a*x + 1))^(1/4)))/(64*a^4) - (475*atan(((a*x - 1)/( a*x + 1))^(1/4)))/(64*a^4) - ((7483*(a*x - 1)^2)/(96*(a*x + 1)^2) - (1805* (a*x - 1)^3)/(32*(a*x + 1)^3) + (475*(a*x - 1)^4)/(32*(a*x + 1)^4) - (4645 *(a*x - 1))/(96*(a*x + 1)) + 8)/(a^4*((a*x - 1)/(a*x + 1))^(1/4) - 4*a^4*( (a*x - 1)/(a*x + 1))^(5/4) + 6*a^4*((a*x - 1)/(a*x + 1))^(9/4) - 4*a^4*((a *x - 1)/(a*x + 1))^(13/4) + a^4*((a*x - 1)/(a*x + 1))^(17/4))