\(\int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx\) [113]

Optimal result

Integrand size = 14, antiderivative size = 250 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=-\frac {2467 \sqrt [4]{1-\frac {1}{a x}}}{192 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {521 \sqrt [4]{1-\frac {1}{a x}} x}{192 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {113 \sqrt [4]{1-\frac {1}{a x}} x^2}{96 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {17 \sqrt [4]{1-\frac {1}{a x}} x^3}{24 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^4}{4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {475 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4}+\frac {475 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^4} \] Output:

-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
 

Mathematica [A] (verified)

Time = 5.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.64 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {-3072 e^{-\frac {1}{2} \coth ^{-1}(a x)}+\frac {1536 e^{\frac {15}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}-\frac {5248 e^{\frac {11}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {7376 e^{\frac {7}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}-\frac {6292 e^{\frac {3}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}+2850 \arctan \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )-1425 \log \left (1-e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )+1425 \log \left (1+e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )}{384 a^4} \] Input:

Integrate[x^3/E^((5*ArcCoth[a*x])/2),x]
 

Output:

(-3072/E^(ArcCoth[a*x]/2) + (1536*E^((15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcC 
oth[a*x]))^4 - (5248*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 
+ (7376*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 - (6292*E^((3* 
ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) + 2850*ArcTan[E^(-1/2*ArcCoth[ 
a*x])] - 1425*Log[1 - E^(-1/2*ArcCoth[a*x])] + 1425*Log[1 + E^(-1/2*ArcCot 
h[a*x])])/(384*a^4)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {6721, 109, 27, 168, 27, 168, 27, 168, 27, 172, 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.

\(\displaystyle \int x^3 e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 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

\(\displaystyle \frac {1}{4} \int \frac {\left (17 a-\frac {16}{x}\right ) x^4}{2 a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (17 a-\frac {16}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {1}{3} \int \frac {\left (113 a-\frac {102}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\left (113 a-\frac {102}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {1}{2} \int \frac {\left (521 a-\frac {452}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (521 a-\frac {452}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {-\int \frac {\left (1425 a-\frac {1042}{x}\right ) x}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {\left (1425 a-\frac {1042}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {-\frac {-\frac {-\frac {2 a \int \frac {1425 x}{2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {1425 a \int \frac {x}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {-\frac {-\frac {-\frac {5700 a \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-5700 a \int \frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {-\frac {-\frac {-\frac {5700 a \left (\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-\frac {-\frac {-\frac {5700 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {-\frac {5700 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4934 a \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{2 a}-\frac {521 a x \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}}{4 a}-\frac {113 a x^2 \sqrt [4]{1-\frac {1}{a x}}}{2 \sqrt [4]{\frac {1}{a x}+1}}}{6 a}-\frac {17 a x^3 \sqrt [4]{1-\frac {1}{a x}}}{3 \sqrt [4]{\frac {1}{a x}+1}}}{8 a^2}+\frac {x^4 \sqrt [4]{1-\frac {1}{a x}}}{4 \sqrt [4]{\frac {1}{a x}+1}}\)

Input:

Int[x^3/E^((5*ArcCoth[a*x])/2),x]
 

Output:

((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*(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))/(2*a))/(4*a))/(6*a))/(8*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

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]
 

rule 109

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])
 

rule 168

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]
 

rule 172

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]
 

rule 216

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])
 

rule 219

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])
 

rule 827

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]
 

rule 6721

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]
 

Maple [F]

\[\int x^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}d x\]

Input:

int(x^3*((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int(x^3*((a*x-1)/(a*x+1))^(5/4),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.44 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {2 \, {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 521 \, a x - 2467\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{384 \, a^{4}} \] Input:

integrate(x^3*((a*x-1)/(a*x+1))^(5/4),x, algorithm="fricas")
 

Output:

1/384*(2*(48*a^4*x^4 - 136*a^3*x^3 + 226*a^2*x^2 - 521*a*x - 2467)*((a*x - 
 1)/(a*x + 1))^(1/4) + 2850*arctan(((a*x - 1)/(a*x + 1))^(1/4)) + 1425*log 
(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 1425*log(((a*x - 1)/(a*x + 1))^(1/4) - 
 1))/a^4
 

Sympy [F]

\[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\int x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}\, dx \] Input:

integrate(x**3*((a*x-1)/(a*x+1))**(5/4),x)
 

Output:

Integral(x**3*((a*x - 1)/(a*x + 1))**(5/4), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=-\frac {1}{384} \, a {\left (\frac {4 \, {\left (1573 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} - 2875 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 2343 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - 657 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{5}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} - \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} + \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{5}} + \frac {3072 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{5}}\right )} \] Input:

integrate(x^3*((a*x-1)/(a*x+1))^(5/4),x, algorithm="maxima")
 

Output:

-1/384*a*(4*(1573*((a*x - 1)/(a*x + 1))^(13/4) - 2875*((a*x - 1)/(a*x + 1) 
)^(9/4) + 2343*((a*x - 1)/(a*x + 1))^(5/4) - 657*((a*x - 1)/(a*x + 1))^(1/ 
4))/(4*(a*x - 1)*a^5/(a*x + 1) - 6*(a*x - 1)^2*a^5/(a*x + 1)^2 + 4*(a*x - 
1)^3*a^5/(a*x + 1)^3 - (a*x - 1)^4*a^5/(a*x + 1)^4 - a^5) - 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(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^5 + 3072*((a*x - 1)/(a* 
x + 1))^(1/4)/a^5)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.89 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {1}{384} \, a {\left (\frac {2850 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{5}} + \frac {1425 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{5}} - \frac {1425 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{5}} - \frac {3072 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{5}} + \frac {4 \, {\left (\frac {2343 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {2875 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {1573 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - 657 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{5} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \] Input:

integrate(x^3*((a*x-1)/(a*x+1))^(5/4),x, algorithm="giac")
 

Output:

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*x - 1)/(a*x + 1))^(1/4)/a^5 + 4*(2343*(a*x - 1)*((a*x - 
1)/(a*x + 1))^(1/4)/(a*x + 1) - 2875*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(1/ 
4)/(a*x + 1)^2 + 1573*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^3 
- 657*((a*x - 1)/(a*x + 1))^(1/4))/(a^5*((a*x - 1)/(a*x + 1) - 1)^4))
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.87 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\frac {475\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{64\,a^4}-\frac {\frac {219\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{32}-\frac {781\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{32}+\frac {2875\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{96}-\frac {1573\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{96}}{a^4+\frac {6\,a^4\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a^4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a^4\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a^4\,\left (a\,x-1\right )}{a\,x+1}}-\frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a^4}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,475{}\mathrm {i}}{64\,a^4} \] Input:

int(x^3*((a*x - 1)/(a*x + 1))^(5/4),x)
 

Output:

(475*atan(((a*x - 1)/(a*x + 1))^(1/4)))/(64*a^4) - (8*((a*x - 1)/(a*x + 1) 
)^(1/4))/a^4 - ((219*((a*x - 1)/(a*x + 1))^(1/4))/32 - (781*((a*x - 1)/(a* 
x + 1))^(5/4))/32 + (2875*((a*x - 1)/(a*x + 1))^(9/4))/96 - (1573*((a*x - 
1)/(a*x + 1))^(13/4))/96)/(a^4 + (6*a^4*(a*x - 1)^2)/(a*x + 1)^2 - (4*a^4* 
(a*x - 1)^3)/(a*x + 1)^3 + (a^4*(a*x - 1)^4)/(a*x + 1)^4 - (4*a^4*(a*x - 1 
))/(a*x + 1)) - (atan(((a*x - 1)/(a*x + 1))^(1/4)*1i)*475i)/(64*a^4)
 

Reduce [F]

\[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^3 \, dx=\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}} x^{4}}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right ) a -\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}} x^{3}}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right ) \] Input:

int(x^3*((a*x-1)/(a*x+1))^(5/4),x)
 

Output:

int(((a*x - 1)**(1/4)*x**4)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x)*a 
 - int(((a*x - 1)**(1/4)*x**3)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x 
)