Integrand size = 22, antiderivative size = 98 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{7 a c^4 (1+a x) \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}} \] Output:
6/35*x/c^4/(-a^2*x^2+1)^(5/2)-1/7/a/c^4/(a*x+1)/(-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 = 0.77 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {-5+30 a x+30 a^2 x^2-40 a^3 x^3-40 a^4 x^4+16 a^5 x^5+16 a^6 x^6}{35 a c^4 (1-a x)^{5/2} (1+a x)^{7/2}} \] Input:
Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^4),x]
Output:
(-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)^(5/2)*(1 + a*x)^(7/2))
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6689, 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.
| \(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6689 |
| \(\displaystyle \frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 454 |
| \(\displaystyle \frac {\frac {6}{7} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}}dx-\frac {1-a x}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )-\frac {1-a x}{7 a \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
Input:
Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^4),x]
Output:
(-1/7*(1 - a*x)/(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
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] && ILtQ[(n - 1)/2, 0] && !In tegerQ[p - n/2]
Time = 0.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76
| method | result | size |
| gosper | \(\frac {16 x^{6} a^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5}{35 a \,c^{4} \left (a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\) | \(74\) |
| trager | \(-\frac {\left (16 x^{6} a^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{4} \left (a x +1\right )^{4} \left (a x -1\right )^{3} a}\) | \(81\) |
| orering | \(-\frac {\left (16 x^{6} a^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right ) \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{35 a \left (-a^{2} c \,x^{2}+c \right )^{4}}\) | \(82\) |
| default | \(\frac {\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{32 a^{4}}-\frac {5 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{192 a^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {\frac {15 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{128 a \left (x -\frac {1}{a}\right )^{2}}+\frac {15 a \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{128}}{a^{2}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{7 a \left (x +\frac {1}{a}\right )^{5}}+\frac {2 a \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}\right )}{7}}{16 a^{5}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}}{8 a^{4}}-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{96 a^{4} \left (x +\frac {1}{a}\right )^{3}}+\frac {-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{32 a \left (x +\frac {1}{a}\right )^{2}}-\frac {5 a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{32}}{a^{2}}+\frac {\frac {35 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{256}+\frac {35 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{256 \sqrt {a^{2}}}}{a}-\frac {35 \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{256 a}}{c^{4}}\) | \(720\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
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. 197 vs. \(2 (82) = 164\).
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.01 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x + {\left (16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1} - 5}{35 \, {\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right )}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm="fric as")
Output:
-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)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**4,x)
Output:
Integral(1/(-a**7*x**7*sqrt(-a**2*x**2 + 1) - a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x *sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**4
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm="maxi ma")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^4*(a*x + 1)), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm="giac ")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^4*(a*x + 1)), x)
Time = 27.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^4}+\frac {1}{56\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {17\,x}{70\,c^4}-\frac {1}{7\,a\,c^4}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{56\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{35\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \] Input:
int((1 - a^2*x^2)^(1/2)/((c - a^2*c*x^2)^4*(a*x + 1)),x)
Output:
((1 - a^2*x^2)^(1/2)*((8*x)/(35*c^4) + 1/(56*a*c^4)))/((a*x - 1)^2*(a*x + 1)^2) - ((1 - a^2*x^2)^(1/2)*((17*x)/(70*c^4) - 1/(7*a*c^4)))/((a*x - 1)^3 *(a*x + 1)^3) - (1 - a^2*x^2)^(1/2)/(56*a*c^4*(a*x + 1)^4) - (16*x*(1 - a^ 2*x^2)^(1/2))/(35*c^4*(a*x - 1)*(a*x + 1))
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {30 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+30 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-60 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-60 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+30 \sqrt {-a^{2} x^{2}+1}\, a x +30 \sqrt {-a^{2} x^{2}+1}+16 a^{6} x^{6}+16 a^{5} x^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5}{35 \sqrt {-a^{2} x^{2}+1}\, a \,c^{4} \left (a^{5} x^{5}+a^{4} x^{4}-2 a^{3} x^{3}-2 a^{2} x^{2}+a x +1\right )} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x)
Output:
(30*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 30*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 60*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 60*sqrt( - a**2*x**2 + 1)*a**2*x* *2 + 30*sqrt( - a**2*x**2 + 1)*a*x + 30*sqrt( - a**2*x**2 + 1) + 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*sqrt( - a**2*x**2 + 1)*a*c**4*(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1))