Integrand size = 14, antiderivative size = 253 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {611 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4}+\frac {269 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3}+\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3}{48 a^2}+\frac {9 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a}+\frac {1}{5} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^5+\frac {31 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {31 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \] Output:
611/1920*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x/a^4+269/960*(1-1/a/x)^(3/4)*(1+ 1/a/x)^(1/4)*x^2/a^3+11/48*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^3/a^2+9/40*(1 -1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^4/a+1/5*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*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 = 5.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.68 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {24576 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {62976 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {64640 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {34000 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {9620 e^{\frac {1}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}+930 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-465 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+465 \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{3840 a^5} \] Input:
Integrate[E^(ArcCoth[a*x]/2)*x^4,x]
Output:
((24576*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^5 + (62976*E^(ArcCot h[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (64640*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (34000*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x] ))^2 + (9620*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 930*ArcTan[E^ (ArcCoth[a*x]/2)] - 465*Log[1 - E^(ArcCoth[a*x]/2)] + 465*Log[1 + E^(ArcCo th[a*x]/2)])/(3840*a^5)
Time = 0.69 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6721, 110, 27, 168, 27, 168, 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.
| \(\displaystyle \int x^4 e^{\frac {1}{2} \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {1}{5} \int \frac {\left (9 a+\frac {8}{x}\right ) x^5}{2 a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\int \frac {\left (9 a+\frac {8}{x}\right ) x^5}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {-\frac {1}{4} \int -\frac {\left (55 a+\frac {54}{x}\right ) x^4}{2 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\int \frac {\left (55 a+\frac {54}{x}\right ) x^4}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {-\frac {1}{3} \int -\frac {\left (269 a+\frac {220}{x}\right ) x^3}{2 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\int \frac {\left (269 a+\frac {220}{x}\right ) x^3}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {-\frac {1}{2} \int -\frac {\left (611 a+\frac {538}{x}\right ) x^2}{2 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {\int \frac {\left (611 a+\frac {538}{x}\right ) x^2}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {-\int -\frac {465 x}{2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {\frac {465}{2} \int \frac {x}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {930 \int \frac {1}{\frac {1}{x^4}-1}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {930 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {930 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} x^5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {\frac {\frac {\frac {930 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )-611 a x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a}-\frac {269}{2} a x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{6 a}-\frac {55}{3} a x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{8 a}-\frac {9}{4} a x^4 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{10 a^2}\) |
Input:
Int[E^(ArcCoth[a*x]/2)*x^4,x]
Output:
((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^5)/5 - ((-9*a*(1 - 1/(a*x))^(3/ 4)*(1 + 1/(a*x))^(1/4)*x^4)/4 + ((-55*a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^ (1/4)*x^3)/3 + ((-269*a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/2 + ( -611*a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x + 930*(-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))/(4*a))/(6*a))/(8*a))/(10*a^2)
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[((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]
\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x)
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.47 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {2 \, {\left (384 \, a^{5} x^{5} + 816 \, a^{4} x^{4} + 872 \, a^{3} x^{3} + 978 \, a^{2} x^{2} + 1149 \, a x + 611\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}} - 930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{3840 \, a^{5}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x, algorithm="fricas")
Output:
1/3840*(2*(384*a^5*x^5 + 816*a^4*x^4 + 872*a^3*x^3 + 978*a^2*x^2 + 1149*a* x + 611)*((a*x - 1)/(a*x + 1))^(3/4) - 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
\[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {x^{4}}{\sqrt [4]{\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/4)*x**4,x)
Output:
Integral(x**4/((a*x - 1)/(a*x + 1))**(1/4), x)
Time = 0.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (465 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {19}{4}} - 696 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{4}} + 5090 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{4}} - 1120 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} + 2405 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac {930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x, algorithm="maxima")
Output:
-1/3840*a*(4*(465*((a*x - 1)/(a*x + 1))^(19/4) - 696*((a*x - 1)/(a*x + 1)) ^(15/4) + 5090*((a*x - 1)/(a*x + 1))^(11/4) - 1120*((a*x - 1)/(a*x + 1))^( 7/4) + 2405*((a*x - 1)/(a*x + 1))^(3/4))/(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 = 0.14 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {930 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {465 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {465 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} - \frac {4 \, {\left (\frac {1120 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {5090 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {696 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {465 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{4}} - 2405 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x, algorithm="giac")
Output:
-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*(1120*(a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 5090*(a* x - 1)^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 696*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^3 - 465*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^( 3/4)/(a*x + 1)^4 - 2405*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))
Time = 0.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {481\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{192}-\frac {7\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{6}+\frac {509\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/4}}{96}-\frac {29\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/4}}{40}+\frac {31\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{19/4}}{64}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}-\frac {31\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}+\frac {31\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5} \] Input:
int(x^4/((a*x - 1)/(a*x + 1))^(1/4),x)
Output:
((481*((a*x - 1)/(a*x + 1))^(3/4))/192 - (7*((a*x - 1)/(a*x + 1))^(7/4))/6 + (509*((a*x - 1)/(a*x + 1))^(11/4))/96 - (29*((a*x - 1)/(a*x + 1))^(15/4 ))/40 + (31*((a*x - 1)/(a*x + 1))^(19/4))/64)/(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)) - (3 1*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)
\[ \int e^{\frac {1}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{4}} x^{4}}{\left (a x -1\right )^{\frac {1}{4}}}d x \] Input:
int(1/((a*x-1)/(a*x+1))^(1/4)*x^4,x)
Output:
int(((a*x + 1)**(1/4)*x**4)/(a*x - 1)**(1/4),x)