Integrand size = 14, antiderivative size = 285 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {26111 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{1920 a^4}+\frac {10289 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{960 a^3}+\frac {455 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3}{48 a^2}+\frac {349 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^4}{40 a}-\frac {8 \sqrt [4]{1+\frac {1}{a x}} x^5}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {41}{5} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^5+\frac {1003 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \] Output:
26111/1920*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x/a^4+10289/960*(1-1/a/x)^(3/4) *(1+1/a/x)^(1/4)*x^2/a^3+455/48*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^3/a^2+34 9/40*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^4/a-8*(1+1/a/x)^(1/4)*x^5/(1-1/a/x) ^(1/4)+41/5*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^5+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
Time = 5.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.69 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {-8 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {32 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {122 e^{\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {233 e^{\frac {1}{2} \coth ^{-1}(a x)}}{6 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {1661 e^{\frac {1}{2} \coth ^{-1}(a x)}}{48 \left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {4117 e^{\frac {1}{2} \coth ^{-1}(a x)}}{192 \left (-1+e^{2 \coth ^{-1}(a x)}\right )}+\frac {1003}{128} \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-\frac {1003}{256} \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+\frac {1003}{256} \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{a^5} \] Input:
Integrate[E^((5*ArcCoth[a*x])/2)*x^4,x]
Output:
(-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*ArcTan[E^(ArcCoth[a*x]/2)])/128 - (1003*Lo g[1 - E^(ArcCoth[a*x]/2)])/256 + (1003*Log[1 + E^(ArcCoth[a*x]/2)])/256)/a ^5
Time = 0.74 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6721, 109, 27, 168, 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.
| \(\displaystyle \int x^4 e^{\frac {5}{2} \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^{5/4} x^6}{\left (1-\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{5} \int -\frac {\left (21 a+\frac {20}{x}\right ) x^5}{2 a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\int \frac {\left (21 a+\frac {20}{x}\right ) x^5}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {-\frac {1}{4} \int -\frac {\left (181 a+\frac {168}{x}\right ) x^4}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\int \frac {\left (181 a+\frac {168}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {-\frac {1}{3} \int -\frac {\left (1189 a+\frac {1086}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\int \frac {\left (1189 a+\frac {1086}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {-\frac {1}{2} \int -\frac {\left (5533 a+\frac {4756}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\int \frac {\left (5533 a+\frac {4756}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {-\int -\frac {\left (15045 a+\frac {11066}{x}\right ) x}{2 a \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {\int \frac {\left (15045 a+\frac {11066}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}-2 a \int -\frac {15045 x}{2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {15045 a \int \frac {x}{\sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}+\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {60180 a \int \frac {1}{\frac {1}{x^4}-1}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {60180 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {60180 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^5 \sqrt [4]{\frac {1}{a x}+1}}{5 \sqrt [4]{1-\frac {1}{a x}}}-\frac {\frac {\frac {\frac {\frac {60180 a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {52222 a \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {5533 a x \sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}}{4 a}-\frac {1189 a x^2 \sqrt [4]{\frac {1}{a x}+1}}{2 \sqrt [4]{1-\frac {1}{a x}}}}{6 a}-\frac {181 a x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}}{8 a}-\frac {21 a x^4 \sqrt [4]{\frac {1}{a x}+1}}{4 \sqrt [4]{1-\frac {1}{a x}}}}{10 a^2}\) |
Input:
Int[E^((5*ArcCoth[a*x])/2)*x^4,x]
Output:
((1 + 1/(a*x))^(1/4)*x^5)/(5*(1 - 1/(a*x))^(1/4)) - ((-21*a*(1 + 1/(a*x))^ (1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) + ((-181*a*(1 + 1/(a*x))^(1/4)*x^3)/(3* (1 - 1/(a*x))^(1/4)) + ((-1189*a*(1 + 1/(a*x))^(1/4)*x^2)/(2*(1 - 1/(a*x)) ^(1/4)) + ((-5533*a*(1 + 1/(a*x))^(1/4)*x)/(1 - 1/(a*x))^(1/4) + ((52222*a *(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4) + 60180*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))/(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[(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]
\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)
Output:
int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.53 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {30090 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 15045 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (384 \, a^{6} x^{6} + 1392 \, a^{5} x^{5} + 2456 \, a^{4} x^{4} + 3826 \, a^{3} x^{3} + 7911 \, a^{2} x^{2} - 20578 \, a x - 26111\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{3840 \, {\left (a^{6} x - a^{5}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="fricas")
Output:
-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 + 38 26*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)
\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\int \frac {x^{4}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(5/4)*x**4,x)
Output:
Integral(x**4/((a*x - 1)/(a*x + 1))**(5/4), x)
Time = 0.11 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.96 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {4 \, {\left (\frac {58985 \, {\left (a x - 1\right )}}{a x + 1} - \frac {125920 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {137930 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {72216 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + \frac {15045 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} - 7680\right )}}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {21}{4}} - 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} + 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} - 10 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 5 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="maxima")
Output:
-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 - 72216*(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)
Time = 0.15 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{3840} \, a {\left (\frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {15045 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} + \frac {30720}{a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {4 \, {\left (\frac {49120 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {61130 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {33816 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {7365 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{4}} - 20585 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="giac")
Output:
-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 + 15045*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 - 20 585*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))
Time = 23.57 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.87 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1003\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {1003\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}-\frac {\frac {787\,{\left (a\,x-1\right )}^2}{6\,{\left (a\,x+1\right )}^2}-\frac {13793\,{\left (a\,x-1\right )}^3}{96\,{\left (a\,x+1\right )}^3}+\frac {3009\,{\left (a\,x-1\right )}^4}{40\,{\left (a\,x+1\right )}^4}-\frac {1003\,{\left (a\,x-1\right )}^5}{64\,{\left (a\,x+1\right )}^5}-\frac {11797\,\left (a\,x-1\right )}{192\,\left (a\,x+1\right )}+8}{a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-10\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}+5\,a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}-a^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{21/4}} \] Input:
int(x^4/((a*x - 1)/(a*x + 1))^(5/4),x)
Output:
(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) - ((787*(a*x - 1)^2)/(6*(a*x + 1)^2) - (137 93*(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) + 1 0*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))
\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx=\left (\int \frac {\left (a x +1\right )^{\frac {1}{4}} x^{5}}{\left (a x -1\right )^{\frac {1}{4}} a x -\left (a x -1\right )^{\frac {1}{4}}}d x \right ) a +\int \frac {\left (a x +1\right )^{\frac {1}{4}} x^{4}}{\left (a x -1\right )^{\frac {1}{4}} a x -\left (a x -1\right )^{\frac {1}{4}}}d x \] Input:
int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)
Output:
int(((a*x + 1)**(1/4)*x**5)/((a*x - 1)**(1/4)*a*x - (a*x - 1)**(1/4)),x)*a + int(((a*x + 1)**(1/4)*x**4)/((a*x - 1)**(1/4)*a*x - (a*x - 1)**(1/4)),x )