Integrand size = 20, antiderivative size = 96 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {1+a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}} \]
1/7*(a*x+1)/a/c^4/(-a^2*x^2+1)^(7/2)+6/35*x/c^4/(-a^2*x^2+1)^(5/2)+8/35*x/ c^4/(-a^2*x^2+1)^(3/2)+16/35*x/c^4/(-a^2*x^2+1)^(1/2)
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {5+30 a x-30 a^2 x^2-40 a^3 x^3+40 a^4 x^4+16 a^5 x^5-16 a^6 x^6}{35 a c^4 (1-a x)^{7/2} (1+a x)^{5/2}} \]
(5 + 30*a*x - 30*a^2*x^2 - 40*a^3*x^3 + 40*a^4*x^4 + 16*a^5*x^5 - 16*a^6*x ^6)/(35*a*c^4*(1 - a*x)^(7/2)*(1 + a*x)^(5/2))
Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6688, 454, 209, 209, 208}
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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6688 |
| \(\displaystyle \frac {\int \frac {a x+1}{\left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 454 |
| \(\displaystyle \frac {\frac {6}{7} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {a x+1}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {a x+1}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {a x+1}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {a x+1}{7 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )}{c^4}\) |
((1 + a*x)/(7*a*(1 - a^2*x^2)^(7/2)) + (6*(x/(5*(1 - a^2*x^2)^(5/2)) + (4* (x/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2])))/5))/7)/c^4
3.10.18.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && !I ntegerQ[p - n/2]
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77
| method | result | size |
| gosper | \(\frac {16 a^{6} x^{6}-16 a^{5} x^{5}-40 a^{4} x^{4}+40 a^{3} x^{3}+30 a^{2} x^{2}-30 a x -5}{35 a \,c^{4} \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\) | \(74\) |
| trager | \(-\frac {\left (16 a^{6} x^{6}-16 a^{5} x^{5}-40 a^{4} x^{4}+40 a^{3} x^{3}+30 a^{2} x^{2}-30 a x -5\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{4} \left (a x -1\right )^{4} \left (a x +1\right )^{3} a}\) | \(81\) |
| default | \(\frac {\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{2}}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{8 a^{4}}-\frac {5 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{32 a^{2} \left (x +\frac {1}{a}\right )}-\frac {3 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{16 a^{3}}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 \left (x -\frac {1}{a}\right )}}{a^{2}}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{5 a \left (x +\frac {1}{a}\right )^{3}}+\frac {2 a \left (-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}\right )}{5}}{16 a^{3}}-\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{32 a^{2} \left (x -\frac {1}{a}\right )}}{c^{4}}\) | \(657\) |
1/35/a/c^4/(a*x-1)/(-a^2*x^2+1)^(5/2)*(16*a^6*x^6-16*a^5*x^5-40*a^4*x^4+40 *a^3*x^3+30*a^2*x^2-30*a*x-5)
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (80) = 160\).
Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.06 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {5 \, a^{7} x^{7} - 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} + 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} - 15 \, a^{2} x^{2} - 5 \, a x - {\left (16 \, a^{6} x^{6} - 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} - 30 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1} + 5}{35 \, {\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}} \]
1/35*(5*a^7*x^7 - 5*a^6*x^6 - 15*a^5*x^5 + 15*a^4*x^4 + 15*a^3*x^3 - 15*a^ 2*x^2 - 5*a*x - (16*a^6*x^6 - 16*a^5*x^5 - 40*a^4*x^4 + 40*a^3*x^3 + 30*a^ 2*x^2 - 30*a*x - 5)*sqrt(-a^2*x^2 + 1) + 5)/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3 *a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\int \frac {a x}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
(Integral(a*x/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x** 2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**8*x**8*sqrt(-a**2*x**2 + 1 ) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{4} \sqrt {-a^{2} x^{2} + 1}} \,d x } \]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{4} \sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Time = 3.50 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.97 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {7\,a\,\sqrt {1-a^2\,x^2}}{80\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{20\,\left (a^4\,c^4\,x^2+2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{140\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{56\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {33\,\sqrt {1-a^2\,x^2}}{160\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {281\,\sqrt {1-a^2\,x^2}}{1120\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}+3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {27\,\sqrt {1-a^2\,x^2}}{560\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \]
(7*a*(1 - a^2*x^2)^(1/2))/(80*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) - (a* (1 - a^2*x^2)^(1/2))/(20*(a^2*c^4 + 2*a^3*c^4*x + a^4*c^4*x^2)) + (a^3*(1 - a^2*x^2)^(1/2))/(140*(a^4*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) + (a*(1 - a^ 2*x^2)^(1/2))/(56*(a^2*c^4 - 4*a^3*c^4*x + 6*a^4*c^4*x^2 - 4*a^5*c^4*x^3 + a^6*c^4*x^4)) + (33*(1 - a^2*x^2)^(1/2))/(160*(-a^2)^(1/2)*(c^4*x*(-a^2)^ (1/2) + (c^4*(-a^2)^(1/2))/a)) + (281*(1 - a^2*x^2)^(1/2))/(1120*(-a^2)^(1 /2)*(c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) + (1 - a^2*x^2)^(1/2)/(80 *(-a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) + (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3*( -a^2)^(1/2) + 3*a*c^4*x^2*(-a^2)^(1/2))) + (27*(1 - a^2*x^2)^(1/2))/(560*( -a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3*(-a ^2)^(1/2) - 3*a*c^4*x^2*(-a^2)^(1/2)))