Integrand size = 22, antiderivative size = 127 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4}+\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )} \]
16/63/a/c^4*((a*x-1)/(a*x+1))^(3/2)+1/9*(2*a*x+1)/a/c^4*((a*x-1)/(a*x+1))^ (3/2)/(-a^2*x^2+1)^3+10/63*(4*a*x+3)/a/c^4*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x ^2+1)^2-8/21*(2*a*x+3)/a/c^4*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x^2+1)
Time = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (19-6 a x-66 a^2 x^2-56 a^3 x^3+24 a^4 x^4+48 a^5 x^5+16 a^6 x^6\right )}{63 c^4 (-1+a x)^2 (1+a x)^5} \]
(Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 6*a*x - 66*a^2*x^2 - 56*a^3*x^3 + 24*a^4*x^ 4 + 48*a^5*x^5 + 16*a^6*x^6))/(63*c^4*(-1 + a*x)^2*(1 + a*x)^5)
Time = 0.58 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6739, 27, 6739, 6739, 6737}
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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 6739 |
\(\displaystyle \frac {10 \int \frac {e^{-3 \coth ^{-1}(a x)}}{c^3 \left (1-a^2 x^2\right )^3}dx}{9 c}+\frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {10 \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-a^2 x^2\right )^3}dx}{9 c^4}+\frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}\) |
\(\Big \downarrow \) 6739 |
\(\displaystyle \frac {10 \left (\frac {12}{7} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-a^2 x^2\right )^2}dx+\frac {(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a \left (1-a^2 x^2\right )^2}\right )}{9 c^4}+\frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}\) |
\(\Big \downarrow \) 6739 |
\(\displaystyle \frac {10 \left (\frac {12}{7} \left (-\frac {2}{5} \int \frac {e^{-3 \coth ^{-1}(a x)}}{1-a^2 x^2}dx-\frac {(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a \left (1-a^2 x^2\right )}\right )+\frac {(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a \left (1-a^2 x^2\right )^2}\right )}{9 c^4}+\frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}\) |
\(\Big \downarrow \) 6737 |
\(\displaystyle \frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 \left (\frac {(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a \left (1-a^2 x^2\right )^2}+\frac {12}{7} \left (\frac {2 e^{-3 \coth ^{-1}(a x)}}{15 a}-\frac {(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a \left (1-a^2 x^2\right )}\right )\right )}{9 c^4}\) |
(1 + 2*a*x)/(9*a*c^4*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^3) + (10*((3 + 4*a*x )/(7*a*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^2) + (12*(2/(15*a*E^(3*ArcCoth[a*x ])) - (3 + 2*a*x)/(5*a*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))))/7))/(9*c^4)
3.7.13.3.1 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[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[ E^(n*ArcCoth[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2))) Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || !IntegerQ[n])
Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a^{2} x^{2}-1\right )^{3} c^{4} a}\) | \(81\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a x -1\right )^{3} c^{4} \left (a x +1\right )^{3} a}\) | \(84\) |
trager | \(\frac {\left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{63 a \,c^{4} \left (a x -1\right )^{2} \left (a x +1\right )^{4}}\) | \(86\) |
1/63*((a*x-1)/(a*x+1))^(3/2)*(16*a^6*x^6+48*a^5*x^5+24*a^4*x^4-56*a^3*x^3- 66*a^2*x^2-6*a*x+19)/(a^2*x^2-1)^3/c^4/a
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
1/63*(16*a^6*x^6 + 48*a^5*x^5 + 24*a^4*x^4 - 56*a^3*x^3 - 66*a^2*x^2 - 6*a *x + 19)*sqrt((a*x - 1)/(a*x + 1))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4* x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \]
(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**9*x**9 + a**8*x**8 - 4*a* *7*x**7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x **2 + a*x + 1), x) + Integral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**9* x**9 + a**8*x**8 - 4*a**7*x**7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x + 1), x))/c**4
Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {1}{4032} \, a {\left (\frac {7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 54 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 189 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 420 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 945 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac {21 \, {\left (\frac {18 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
1/4032*a*((7*((a*x - 1)/(a*x + 1))^(9/2) - 54*((a*x - 1)/(a*x + 1))^(7/2) + 189*((a*x - 1)/(a*x + 1))^(5/2) - 420*((a*x - 1)/(a*x + 1))^(3/2) + 945* sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + 21*(18*(a*x - 1)/(a*x + 1) - 1)/(a^ 2*c^4*((a*x - 1)/(a*x + 1))^(3/2)))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]
Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{64\,a\,c^4}-\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\frac {6\,\left (a\,x-1\right )}{a\,x+1}-\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
(15*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - (5*((a*x - 1)/(a*x + 1))^(3/ 2))/(48*a*c^4) + (3*((a*x - 1)/(a*x + 1))^(5/2))/(64*a*c^4) - (3*((a*x - 1 )/(a*x + 1))^(7/2))/(224*a*c^4) + ((a*x - 1)/(a*x + 1))^(9/2)/(576*a*c^4) + ((6*(a*x - 1))/(a*x + 1) - 1/3)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(3/2))