Integrand size = 19, antiderivative size = 155 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}-\frac {5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \]
16/7*a*(a*x+1)/c^4/(-a^2*x^2+1)^(7/2)+4/35*a*(17*a*x+7)/c^4/(-a^2*x^2+1)^( 5/2)+1/105*a*(307*a*x+175)/c^4/(-a^2*x^2+1)^(3/2)-5*a*arctanh((-a^2*x^2+1) ^(1/2))/c^4+1/105*a*(719*a*x+525)/c^4/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2 )/c^4/x
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\frac {105-1339 a x+1812 a^2 x^2+485 a^3 x^3-1947 a^4 x^4+824 a^5 x^5-525 a x (-1+a x)^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{105 c^4 x (-1+a x)^3 \sqrt {1-a^2 x^2}} \]
(105 - 1339*a*x + 1812*a^2*x^2 + 485*a^3*x^3 - 1947*a^4*x^4 + 824*a^5*x^5 - 525*a*x*(-1 + a*x)^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(105* c^4*x*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])
Time = 0.69 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {6678, 27, 570, 532, 25, 2336, 25, 2336, 25, 2336, 27, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^5 x^2 (1-a x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^2 (1-a x)^5}dx}{c^4}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^5}{x^2 \left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int -\frac {-7 a^3 x^3+61 a^2 x^2+35 a x+7}{x^2 \left (1-a^2 x^2\right )^{7/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {-7 a^3 x^3+61 a^2 x^2+35 a x+7}{x^2 \left (1-a^2 x^2\right )^{7/2}}dx+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {272 a^2 x^2+175 a x+35}{x^2 \left (1-a^2 x^2\right )^{5/2}}dx\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {272 a^2 x^2+175 a x+35}{x^2 \left (1-a^2 x^2\right )^{5/2}}dx+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {614 a^2 x^2+525 a x+105}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {614 a^2 x^2+525 a x+105}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}-\int -\frac {105 (5 a x+1)}{x^2 \sqrt {1-a^2 x^2}}dx\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \int \frac {5 a x+1}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {5}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (-5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (719 a x+525)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (307 a x+175)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a (17 a x+7)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
((16*a*(1 + a*x))/(7*(1 - a^2*x^2)^(7/2)) + ((4*a*(7 + 17*a*x))/(5*(1 - a^ 2*x^2)^(5/2)) + ((a*(175 + 307*a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a*(525 + 719*a*x))/Sqrt[1 - a^2*x^2] + 105*(-(Sqrt[1 - a^2*x^2]/x) - 5*a*ArcTanh[Sq rt[1 - a^2*x^2]]))/3)/5)/7)/c^4
3.4.61.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 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_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(422\) vs. \(2(135)=270\).
Time = 0.24 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.73
| method | result | size |
| default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-5 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {3 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{2}}+\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{c^{4}}\) | \(423\) |
| risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c^{4}}-\frac {a \left (5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {4 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{a}+\frac {\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{2}}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}-\frac {2 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}\right )}{a^{3}}\right )}{c^{4}}\) | \(483\) |
1/c^4*(-(-a^2*x^2+1)^(1/2)/x-5*a*arctanh(1/(-a^2*x^2+1)^(1/2))-3/a*(1/5/a/ (x-1/a)^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1 /a)^2*a^2-2*(x-1/a)*a)^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2 )))+2/a^2*(1/7/a/(x-1/a)^4*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-3/7*a*(1/5/a /(x-1/a)^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x- 1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/ 2))))+4/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-19/3/(x-1/a)*(-(x -1/a)^2*a^2-2*(x-1/a)*a)^(1/2))
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\frac {1024 \, a^{5} x^{5} - 4096 \, a^{4} x^{4} + 6144 \, a^{3} x^{3} - 4096 \, a^{2} x^{2} + 1024 \, a x + 525 \, {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (824 \, a^{4} x^{4} - 2771 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 1444 \, a x + 105\right )} \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{4} x^{5} - 4 \, a^{3} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{3} - 4 \, a c^{4} x^{2} + c^{4} x\right )}} \]
1/105*(1024*a^5*x^5 - 4096*a^4*x^4 + 6144*a^3*x^3 - 4096*a^2*x^2 + 1024*a* x + 525*(a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2 + a*x)*log((sqrt(-a^2 *x^2 + 1) - 1)/x) - (824*a^4*x^4 - 2771*a^3*x^3 + 3256*a^2*x^2 - 1444*a*x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^5 - 4*a^3*c^4*x^4 + 6*a^2*c^4*x^3 - 4*a*c^4*x^2 + c^4*x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\frac {\int \frac {a x}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
(Integral(a*x/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 4*a**3*x**5*sqrt(-a**2*x** 2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2 + 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 4*a**3*x**5*sqrt(-a**2*x**2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2 + 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c**4
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (135) = 270\).
Time = 0.29 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.08 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=-\frac {5 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{4} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, c^{4} x {\left | a \right |}} - \frac {{\left (105 \, a^{2} - \frac {4831 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x} + \frac {24997 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{2} x^{2}} - \frac {61131 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{4} x^{3}} + \frac {82915 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{6} x^{4}} - \frac {66325 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{8} x^{5}} + \frac {29295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{10} x^{6}} - \frac {5985 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{12} x^{7}}\right )} a^{2} x}{210 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
-5*a^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c^4* abs(a)) - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(c^4*x*abs(a)) - 1/210*(105* a^2 - 4831*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/x + 24997*(sqrt(-a^2*x^2 + 1)*a bs(a) + a)^2/(a^2*x^2) - 61131*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^4*x^3) + 82915*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^6*x^4) - 66325*(sqrt(-a^2*x^ 2 + 1)*abs(a) + a)^5/(a^8*x^5) + 29295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/( a^10*x^6) - 5985*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^12*x^7))*a^2*x/((sqr t(-a^2*x^2 + 1)*abs(a) + a)*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
Time = 3.51 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a c x)^4} \, dx=\frac {26\,a^3\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^5\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^4\,x}+\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {719\,a^2\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {27\,a^2\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{c^4} \]
(26*a^3*(1 - a^2*x^2)^(1/2))/(15*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) + (4*a^5*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) - ( 1 - a^2*x^2)^(1/2)/(c^4*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*5i)/c^4 + (2* a^3*(1 - a^2*x^2)^(1/2))/(7*(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)) + (719*a^2*(1 - a^2*x^2)^(1/2))/(105*(-a^2)^(1/2) *(c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) + (27*a^2*(1 - a^2*x^2)^(1/2 ))/(35*(-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)))