Integrand size = 18, antiderivative size = 125 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=-\frac {46 \left (a+\frac {1}{x}\right )}{35 a^2 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {35 a+\frac {13}{x}}{35 a^2 c^6 \sqrt {1-\frac {1}{a^2 x^2}}} \]
-46/35*(a+1/x)/a^2/c^6/(1-1/a^2/x^2)^(3/2)+24/35*(a+1/x)^2/a^3/c^6/(1-1/a^ 2/x^2)^(5/2)-1/7*(a+1/x)^3/a^4/c^6/(1-1/a^2/x^2)^(7/2)+1/35*(35*a+13/x)/a^ 2/c^6/(1-1/a^2/x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-13+4 a x+20 a^2 x^2-24 a^3 x^3+8 a^4 x^4\right )}{35 c^6 (-1+a x)^4 (1+a x)} \]
(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^ 4))/(35*c^6*(-1 + a*x)^4*(1 + a*x))
Time = 0.57 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6724, 25, 27, 570, 529, 2166, 2166, 27, 453}
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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle \frac {\int -\frac {1}{c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^3 x^4}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {1}{c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^3 x^4}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^3 x^4}d\frac {1}{x}}{a^3 c^6}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^3}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^4}d\frac {1}{x}}{a^9 c^6}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \int \frac {\left (a+\frac {1}{x}\right )^2 \left (3 a^4+\frac {7 a^3}{x}+\frac {7 a^2}{x^2}+\frac {7 a}{x^3}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^9 c^6}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \int \frac {\left (a+\frac {1}{x}\right ) \left (33 a^4+\frac {70 a^3}{x}+\frac {35 a^2}{x^2}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}\right )}{a^9 c^6}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} a \int \frac {3 a^3 \left (13 a+\frac {35}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )\right )}{a^9 c^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-a^4 \int \frac {13 a+\frac {35}{x}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )\right )}{a^9 c^6}\) |
| \(\Big \downarrow \) 453 |
| \(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {a^5 \left (35 a+\frac {13}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )\right )}{a^9 c^6}\) |
-((-1/7*(a*(-1/5*(a*((46*a^5*(a + x^(-1)))/(1 - 1/(a^2*x^2))^(3/2) - (a^5* (35*a + 13/x))/Sqrt[1 - 1/(a^2*x^2)])) + (24*a^5*(a + x^(-1))^2)/(5*(1 - 1 /(a^2*x^2))^(5/2)))) + (a^5*(a + x^(-1))^3)/(7*(1 - 1/(a^2*x^2))^(7/2)))/( a^9*c^6))
3.3.25.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^n Subst[Int[(d + c*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(p + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0] && IntegerQ[p] && In tegerQ[n]
Time = 0.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.50
| method | result | size |
| trager | \(\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{35 a \,c^{6} \left (a x -1\right )^{4}}\) | \(63\) |
| gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{5} c^{6} a}\) | \(66\) |
| default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -13\right )}{35 \left (a x -1\right )^{5} c^{6} a}\) | \(69\) |
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{6} x^{4} - 4 \, a^{4} c^{6} x^{3} + 6 \, a^{3} c^{6} x^{2} - 4 \, a^{2} c^{6} x + a c^{6}\right )}} \]
1/35*(8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*x - 13)*sqrt((a*x - 1)/(a* x + 1))/(a^5*c^6*x^4 - 4*a^4*c^6*x^3 + 6*a^3*c^6*x^2 - 4*a^2*c^6*x + a*c^6 )
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\, dx}{c^{6}} \]
(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9* a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + 9*a**2*x**2 - 5*a*x + 1), x) + Int egral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9*a **5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + 9*a**2*x**2 - 5*a*x + 1), x))/c**6
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {1}{560} \, a {\left (\frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{6}} + \frac {\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \]
1/560*a*(35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^6) + (28*(a*x - 1)/(a*x + 1) - 70*(a*x - 1)^2/(a*x + 1)^2 + 140*(a*x - 1)^3/(a*x + 1)^3 - 5)/(a^2*c^6*( (a*x - 1)/(a*x + 1))^(7/2)))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a c x - c\right )}^{6}} \,d x } \]
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {8\,a^4\,x^4-24\,a^3\,x^3+20\,a^2\,x^2+4\,a\,x-13}{35\,a\,c^6\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]