Integrand size = 12, antiderivative size = 121 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=-7 a^4 \sqrt {1-\frac {1}{a^2 x^2}}+a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {4 a^3 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {19 a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{8 x}-\frac {51}{8} a^4 \csc ^{-1}(a x) \] Output:
-7*a^4*(1-1/a^2/x^2)^(1/2)+a^4*(1-1/a^2/x^2)^(3/2)-4*a^3*(a-1/x)/(1-1/a^2/ x^2)^(1/2)+1/4*a*(1-1/a^2/x^2)^(1/2)/x^3+19/8*a^3*(1-1/a^2/x^2)^(1/2)/x-51 /8*a^4*arccsc(a*x)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (-2+6 a x-11 a^2 x^2+29 a^3 x^3+80 a^4 x^4\right )}{8 x^3 (1+a x)}-\frac {51}{8} a^4 \arcsin \left (\frac {1}{a x}\right ) \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
-1/8*(a*Sqrt[1 - 1/(a^2*x^2)]*(-2 + 6*a*x - 11*a^2*x^2 + 29*a^3*x^3 + 80*a ^4*x^4))/(x^3*(1 + a*x)) - (51*a^4*ArcSin[1/(a*x)])/8
Time = 1.70 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6719, 2164, 27, 2027, 2164, 27, 563, 25, 2346, 25, 2346, 27, 2346, 27, 455, 223}
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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6719 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1+\frac {1}{a x}\right ) x^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle -\frac {\int \frac {a^2 \left (\frac {a}{x^3}-\frac {1}{x^4}\right ) \sqrt {1-\frac {1}{a^2 x^2}}}{\left (a+\frac {1}{x}\right )^2}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a \int \frac {\left (\frac {a}{x^3}-\frac {1}{x^4}\right ) \sqrt {1-\frac {1}{a^2 x^2}}}{\left (a+\frac {1}{x}\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}{\left (a+\frac {1}{x}\right )^2 x^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle -a^2 \int \frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a+\frac {1}{x}\right )^3 x^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a+\frac {1}{x}\right )^3 x^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 563 |
| \(\displaystyle -a^3 \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\frac {\int -\frac {4 a^4-\frac {4 a^3}{x}+\frac {4 a^2}{x^2}-\frac {3 a}{x^3}+\frac {1}{x^4}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 \left (\frac {\int \frac {4 a^4-\frac {4 a^3}{x}+\frac {4 a^2}{x^2}-\frac {3 a}{x^3}+\frac {1}{x^4}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {-\frac {1}{4} a^2 \int -\frac {16 a^2-\frac {16 a}{x}+\frac {19}{x^2}-\frac {12}{x^3 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \int \frac {16 a^2-\frac {16 a}{x}+\frac {19}{x^2}-\frac {12}{x^3 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {1}{3} a^2 \int -\frac {3 \left (16-\frac {24}{a x}+\frac {19}{a^2 x^2}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \left (a^2 \int \frac {16-\frac {24}{a x}+\frac {19}{a^2 x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \left (a^2 \left (-\frac {1}{2} a^2 \int -\frac {3 \left (17 a-\frac {16}{x}\right )}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \left (a^2 \left (\frac {3 \int \frac {17 a-\frac {16}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -a^3 \left (\frac {\frac {1}{4} a^2 \left (a^2 \left (\frac {3 \left (17 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -a^3 \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+\frac {\frac {1}{4} a^2 \left (a^2 \left (\frac {3 \left (17 a^2 \arcsin \left (\frac {1}{a x}\right )+16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}}{a^4}\right )\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
-(a^3*((4*a^2*Sqrt[1 - 1/(a^2*x^2)])/(a + x^(-1)) + (-1/4*(a^2*Sqrt[1 - 1/ (a^2*x^2)])/x^3 + (a^2*((4*a*Sqrt[1 - 1/(a^2*x^2)])/x^2 + a^2*((-19*Sqrt[1 - 1/(a^2*x^2)])/(2*x) + (3*(16*a^2*Sqrt[1 - 1/(a^2*x^2)] + 17*a^2*ArcSin[ 1/(a*x)]))/(2*a))))/4)/a^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*e Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* (a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 ]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x , 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(-\frac {\left (a x +1\right ) \left (48 a^{3} x^{3}-19 a^{2} x^{2}+8 a x -2\right ) \sqrt {\frac {a x -1}{a x +1}}}{8 x^{4}}+\frac {\left (-\frac {51 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}-\frac {4 a^{3} \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(137\) |
| default | \(\frac {\left (-56 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{7} x^{7}+56 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-163 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}-51 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{6} x^{6}+56 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+56 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{6}-56 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+91 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-158 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}-102 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{5} x^{5}+112 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{4} x^{4}+112 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-112 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+22 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}-51 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-51 a^{4} \sqrt {a^{2}}\, x^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+56 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+56 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-56 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-7 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+4 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 \sqrt {a^{2}}\, x^{4} \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(690\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/8*(a*x+1)*(48*a^3*x^3-19*a^2*x^2+8*a*x-2)/x^4*((a*x-1)/(a*x+1))^(1/2)+( -51/8*a^4*arctan(1/(a^2*x^2-1)^(1/2))-4*a^3/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+ 1/a))^(1/2))*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.64 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\frac {102 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, x^{4}} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="fricas")
Output:
1/8*(102*a^4*x^4*arctan(sqrt((a*x - 1)/(a*x + 1))) - (80*a^4*x^4 + 29*a^3* x^3 - 11*a^2*x^2 + 6*a*x - 2)*sqrt((a*x - 1)/(a*x + 1)))/x^4
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/x**5,x)
Output:
Integral(((a*x - 1)/(a*x + 1))**(3/2)/x**5, x)
Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.60 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{4} \, {\left (51 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 16 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {77 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 149 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 123 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 35 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="maxima")
Output:
1/4*(51*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - 16*a^3*sqrt((a*x - 1)/(a*x + 1)) - (77*a^3*((a*x - 1)/(a*x + 1))^(7/2) + 149*a^3*((a*x - 1)/(a*x + 1 ))^(5/2) + 123*a^3*((a*x - 1)/(a*x + 1))^(3/2) + 35*a^3*sqrt((a*x - 1)/(a* x + 1)))/(4*(a*x - 1)/(a*x + 1) + 6*(a*x - 1)^2/(a*x + 1)^2 + 4*(a*x - 1)^ 3/(a*x + 1)^3 + (a*x - 1)^4/(a*x + 1)^4 + 1))*a
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="giac")
Output:
undef
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\frac {51\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}-4\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {\frac {35\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {123\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {149\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}+\frac {77\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {4\,\left (a\,x-1\right )}{a\,x+1}+1} \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/x^5,x)
Output:
(51*a^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/4 - 4*a^4*((a*x - 1)/(a*x + 1)) ^(1/2) - ((35*a^4*((a*x - 1)/(a*x + 1))^(1/2))/4 + (123*a^4*((a*x - 1)/(a* x + 1))^(3/2))/4 + (149*a^4*((a*x - 1)/(a*x + 1))^(5/2))/4 + (77*a^4*((a*x - 1)/(a*x + 1))^(7/2))/4)/((6*(a*x - 1)^2)/(a*x + 1)^2 + (4*(a*x - 1)^3)/ (a*x + 1)^3 + (a*x - 1)^4/(a*x + 1)^4 + (4*(a*x - 1))/(a*x + 1) + 1)
Time = 0.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx=\frac {102 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{5} x^{5}+102 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{4} x^{4}-102 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{5} x^{5}-102 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{4} x^{4}-80 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}-29 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+11 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-6 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \sqrt {a x +1}\, \sqrt {a x -1}-22 a^{5} x^{5}-22 a^{4} x^{4}}{8 x^{4} \left (a x +1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/x^5,x)
Output:
(102*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**5*x**5 + 102*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**4*x**4 - 102*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**5*x**5 - 102*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**4*x**4 - 80*sqrt(a*x + 1)*sqrt(a*x - 1)*a**4*x**4 - 29*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 + 11*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 6*sqrt(a*x + 1)* sqrt(a*x - 1)*a*x + 2*sqrt(a*x + 1)*sqrt(a*x - 1) - 22*a**5*x**5 - 22*a**4 *x**4)/(8*x**4*(a*x + 1))