\(\int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx\) [77]

Optimal result

Integrand size = 14, antiderivative size = 287 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \]

[Out]

-26111/1920*(1+1/a/x)^(1/4)/a^5/(1-1/a/x)^(1/4)+5533/1920*(1+1/a/x)^(1/4)*x/a^4/(1-1/a/x)^(1/4)+1189/960*(1+1/
a/x)^(1/4)*x^2/a^3/(1-1/a/x)^(1/4)+181/240*(1+1/a/x)^(1/4)*x^3/a^2/(1-1/a/x)^(1/4)+21/40*(1+1/a/x)^(1/4)*x^4/a
/(1-1/a/x)^(1/4)+1/5*(1+1/a/x)^(1/4)*x^5/(1-1/a/x)^(1/4)+1003/128*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5+
1003/128*arctanh((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6306, 100, 156, 160, 12, 95, 218, 212, 209} \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1003 \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {26111 \sqrt [4]{\frac {1}{a x}+1}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 x \sqrt [4]{\frac {1}{a x}+1}}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 x^2 \sqrt [4]{\frac {1}{a x}+1}}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 x^3 \sqrt [4]{\frac {1}{a x}+1}}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 x^4 \sqrt [4]{\frac {1}{a x}+1}}{40 a \sqrt [4]{1-\frac {1}{a x}}} \]

[In]

Int[E^((5*ArcCoth[a*x])/2)*x^4,x]

[Out]

(-26111*(1 + 1/(a*x))^(1/4))/(1920*a^5*(1 - 1/(a*x))^(1/4)) + (5533*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4*(1 - 1/(a
*x))^(1/4)) + (1189*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 - 1/(a*x))^(1/4)) + (181*(1 + 1/(a*x))^(1/4)*x^3)/(24
0*a^2*(1 - 1/(a*x))^(1/4)) + (21*(1 + 1/(a*x))^(1/4)*x^4)/(40*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^
5)/(5*(1 - 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (1003*ArcTanh[
(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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]

Rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 6306

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{x^6 \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{5} \text {Subst}\left (\int \frac {-\frac {21}{2 a}-\frac {10 x}{a^2}}{x^5 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{20} \text {Subst}\left (\int \frac {\frac {181}{4 a^2}+\frac {42 x}{a^3}}{x^4 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{60} \text {Subst}\left (\int \frac {-\frac {1189}{8 a^3}-\frac {543 x}{4 a^4}}{x^3 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{120} \text {Subst}\left (\int \frac {\frac {5533}{16 a^4}+\frac {1189 x}{4 a^5}}{x^2 \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{120} \text {Subst}\left (\int \frac {-\frac {15045}{32 a^5}-\frac {5533 x}{16 a^6}}{x \left (1-\frac {x}{a}\right )^{5/4} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{60} a \text {Subst}\left (\int \frac {15045}{64 a^6 x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1003 \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{256 a^5} \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {1003 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^5} \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \\ & = -\frac {26111 \sqrt [4]{1+\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1189 \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {181 \sqrt [4]{1+\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {21 \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}} x^5}{5 \sqrt [4]{1-\frac {1}{a x}}}+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.18 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.69 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {-8 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {32 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {122 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {233 e^{\frac {1}{2} \coth ^{-1}(a x)}}{6 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {1661 e^{\frac {1}{2} \coth ^{-1}(a x)}}{48 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {4117 e^{\frac {1}{2} \coth ^{-1}(a x)}}{192 \left (-1+e^{2 \coth ^{-1}(a x)}\right )}+\frac {1003}{128} \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-\frac {1003}{256} \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+\frac {1003}{256} \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{a^5} \]

[In]

Integrate[E^((5*ArcCoth[a*x])/2)*x^4,x]

[Out]

(-8*E^(ArcCoth[a*x]/2) + (32*E^(ArcCoth[a*x]/2))/(5*(-1 + E^(2*ArcCoth[a*x]))^5) + (122*E^(ArcCoth[a*x]/2))/(5
*(-1 + E^(2*ArcCoth[a*x]))^4) + (233*E^(ArcCoth[a*x]/2))/(6*(-1 + E^(2*ArcCoth[a*x]))^3) + (1661*E^(ArcCoth[a*
x]/2))/(48*(-1 + E^(2*ArcCoth[a*x]))^2) + (4117*E^(ArcCoth[a*x]/2))/(192*(-1 + E^(2*ArcCoth[a*x]))) + (1003*Ar
cTan[E^(ArcCoth[a*x]/2)])/128 - (1003*Log[1 - E^(ArcCoth[a*x]/2)])/256 + (1003*Log[1 + E^(ArcCoth[a*x]/2)])/25
6)/a^5

Maple [F]

\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]

[In]

int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.53 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {30090 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (384 \, a^{6} x^{6} + 1392 \, a^{5} x^{5} + 2456 \, a^{4} x^{4} + 3826 \, a^{3} x^{3} + 7911 \, a^{2} x^{2} - 20578 \, a x - 26111\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{3840 \, {\left (a^{6} x - a^{5}\right )}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="fricas")

[Out]

-1/3840*(30090*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 15045*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4)
+ 1) + 15045*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) - 1) - 2*(384*a^6*x^6 + 1392*a^5*x^5 + 2456*a^4*x^4 +
3826*a^3*x^3 + 7911*a^2*x^2 - 20578*a*x - 26111)*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*x - a^5)

Sympy [F]

\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {x^{4}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(5/4)*x**4,x)

[Out]

Integral(x**4/((a*x - 1)/(a*x + 1))**(5/4), x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.96 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (\frac {58985 \, {\left (a x - 1\right )}}{a x + 1} - \frac {125920 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {137930 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {72216 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + \frac {15045 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} - 7680\right )}}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {21}{4}} - 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} + 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} - 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="maxima")

[Out]

-1/3840*a*(4*(58985*(a*x - 1)/(a*x + 1) - 125920*(a*x - 1)^2/(a*x + 1)^2 + 137930*(a*x - 1)^3/(a*x + 1)^3 - 72
216*(a*x - 1)^4/(a*x + 1)^4 + 15045*(a*x - 1)^5/(a*x + 1)^5 - 7680)/(a^6*((a*x - 1)/(a*x + 1))^(21/4) - 5*a^6*
((a*x - 1)/(a*x + 1))^(17/4) + 10*a^6*((a*x - 1)/(a*x + 1))^(13/4) - 10*a^6*((a*x - 1)/(a*x + 1))^(9/4) + 5*a^
6*((a*x - 1)/(a*x + 1))^(5/4) - a^6*((a*x - 1)/(a*x + 1))^(1/4)) + 30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a
^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 + 15045*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} + \frac {30720}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {4 \, {\left (\frac {49120 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {61130 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {33816 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {7365 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{4}} - 20585 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="giac")

[Out]

-1/3840*a*(30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 + 15
045*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^6 + 30720/(a^6*((a*x - 1)/(a*x + 1))^(1/4)) - 4*(49120*(a*x -
1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 61130*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 33816*(
a*x - 1)^3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^3 - 7365*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^4
- 20585*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))

Mupad [B] (verification not implemented)

Time = 4.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1003\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {1003\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {\frac {787\,{\left (a\,x-1\right )}^2}{6\,{\left (a\,x+1\right )}^2}-\frac {13793\,{\left (a\,x-1\right )}^3}{96\,{\left (a\,x+1\right )}^3}+\frac {3009\,{\left (a\,x-1\right )}^4}{40\,{\left (a\,x+1\right )}^4}-\frac {1003\,{\left (a\,x-1\right )}^5}{64\,{\left (a\,x+1\right )}^5}-\frac {11797\,\left (a\,x-1\right )}{192\,\left (a\,x+1\right )}+8}{a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}+5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}-a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{21/4}} \]

[In]

int(x^4/((a*x - 1)/(a*x + 1))^(5/4),x)

[Out]

(1003*atanh(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5) - (1003*atan(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5) - ((7
87*(a*x - 1)^2)/(6*(a*x + 1)^2) - (13793*(a*x - 1)^3)/(96*(a*x + 1)^3) + (3009*(a*x - 1)^4)/(40*(a*x + 1)^4) -
(1003*(a*x - 1)^5)/(64*(a*x + 1)^5) - (11797*(a*x - 1))/(192*(a*x + 1)) + 8)/(a^5*((a*x - 1)/(a*x + 1))^(1/4)
- 5*a^5*((a*x - 1)/(a*x + 1))^(5/4) + 10*a^5*((a*x - 1)/(a*x + 1))^(9/4) - 10*a^5*((a*x - 1)/(a*x + 1))^(13/4
) + 5*a^5*((a*x - 1)/(a*x + 1))^(17/4) - a^5*((a*x - 1)/(a*x + 1))^(21/4))