Integrand size = 19, antiderivative size = 192 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\frac {16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (1470+1867 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^4 x^2}-\frac {5 a \sqrt {1-a^2 x^2}}{c^4 x}-\frac {29 a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c^4} \]
16/7*a^2*(a*x+1)/c^4/(-a^2*x^2+1)^(7/2)+4/35*a^2*(31*a*x+21)/c^4/(-a^2*x^2 +1)^(5/2)+1/105*a^2*(671*a*x+455)/c^4/(-a^2*x^2+1)^(3/2)-29/2*a^2*arctanh( (-a^2*x^2+1)^(1/2))/c^4+1/105*a^2*(1867*a*x+1470)/c^4/(-a^2*x^2+1)^(1/2)-1 /2*(-a^2*x^2+1)^(1/2)/c^4/x^2-5*a*(-a^2*x^2+1)^(1/2)/c^4/x
Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\frac {105+735 a x-7774 a^2 x^2+10512 a^3 x^3+2825 a^4 x^4-11307 a^5 x^5+4784 a^6 x^6-3045 a^2 x^2 (-1+a x)^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{210 c^4 x^2 (-1+a x)^3 \sqrt {1-a^2 x^2}} \]
(105 + 735*a*x - 7774*a^2*x^2 + 10512*a^3*x^3 + 2825*a^4*x^4 - 11307*a^5*x ^5 + 4784*a^6*x^6 - 3045*a^2*x^2*(-1 + a*x)^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sq rt[1 - a^2*x^2]])/(210*c^4*x^2*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])
Time = 0.82 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.947, Rules used = {6678, 27, 570, 532, 25, 2336, 25, 2336, 25, 2336, 27, 2338, 25, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^5 x^3 (1-a x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^3 (1-a x)^5}dx}{c^4}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^5}{x^3 \left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int -\frac {89 a^3 x^3+77 a^2 x^2+35 a x+7}{x^3 \left (1-a^2 x^2\right )^{7/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {89 a^3 x^3+77 a^2 x^2+35 a x+7}{x^3 \left (1-a^2 x^2\right )^{7/2}}dx+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {496 a^3 x^3+420 a^2 x^2+175 a x+35}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {496 a^3 x^3+420 a^2 x^2+175 a x+35}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {1342 a^3 x^3+1365 a^2 x^2+525 a x+105}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {1342 a^3 x^3+1365 a^2 x^2+525 a x+105}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}-\int -\frac {105 \left (14 a^2 x^2+5 a x+1\right )}{x^3 \sqrt {1-a^2 x^2}}dx\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \int \frac {14 a^2 x^2+5 a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (-\frac {1}{2} \int -\frac {a (29 a x+10)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} \int \frac {a (29 a x+10)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} a \int \frac {29 a x+10}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} a \left (29 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} a \left (\frac {29}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} a \left (-\frac {29 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \left (\frac {1}{2} a \left (-29 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (1867 a x+1470)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (671 a x+455)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {4 a^2 (31 a x+21)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 a^2 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
((16*a^2*(1 + a*x))/(7*(1 - a^2*x^2)^(7/2)) + ((4*a^2*(21 + 31*a*x))/(5*(1 - a^2*x^2)^(5/2)) + ((a^2*(455 + 671*a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a^ 2*(1470 + 1867*a*x))/Sqrt[1 - a^2*x^2] + 105*(-1/2*Sqrt[1 - a^2*x^2]/x^2 + (a*((-10*Sqrt[1 - a^2*x^2])/x - 29*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3)/ 5)/7)/c^4
3.4.62.3.1 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[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
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. \(396\) vs. \(2(166)=332\).
Time = 0.27 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.07
| method | result | size |
| default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {29 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {5 a \sqrt {-a^{2} x^{2}+1}}{x}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )^{3}}+11 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a}-\frac {14 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}}{c^{4}}\) | \(397\) |
| risch | \(\frac {10 a^{3} x^{3}+a^{2} x^{2}-10 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c^{4}}+\frac {a^{2} \left (-29 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {12 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{3}}+\frac {\frac {6 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )^{2}}-\frac {6 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}}{a}-\frac {10 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a^{2}}-\frac {28 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}\right )}{2 c^{4}}\) | \(498\) |
1/c^4*(-1/2*(-a^2*x^2+1)^(1/2)/x^2-29/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))- 5*a*(-a^2*x^2+1)^(1/2)/x-1/a/(x-1/a)^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)+ 11*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*(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))))-14*a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^( 1/2))
Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\frac {4834 \, a^{6} x^{6} - 19336 \, a^{5} x^{5} + 29004 \, a^{4} x^{4} - 19336 \, a^{3} x^{3} + 4834 \, a^{2} x^{2} + 3045 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 6 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (4784 \, a^{5} x^{5} - 16091 \, a^{4} x^{4} + 18916 \, a^{3} x^{3} - 8404 \, a^{2} x^{2} + 630 \, a x + 105\right )} \sqrt {-a^{2} x^{2} + 1}}{210 \, {\left (a^{4} c^{4} x^{6} - 4 \, a^{3} c^{4} x^{5} + 6 \, a^{2} c^{4} x^{4} - 4 \, a c^{4} x^{3} + c^{4} x^{2}\right )}} \]
1/210*(4834*a^6*x^6 - 19336*a^5*x^5 + 29004*a^4*x^4 - 19336*a^3*x^3 + 4834 *a^2*x^2 + 3045*(a^6*x^6 - 4*a^5*x^5 + 6*a^4*x^4 - 4*a^3*x^3 + a^2*x^2)*lo g((sqrt(-a^2*x^2 + 1) - 1)/x) - (4784*a^5*x^5 - 16091*a^4*x^4 + 18916*a^3* x^3 - 8404*a^2*x^2 + 630*a*x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^6 - 4*a ^3*c^4*x^5 + 6*a^2*c^4*x^4 - 4*a*c^4*x^3 + c^4*x^2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\frac {\int \frac {a x}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
(Integral(a*x/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 4*a**3*x**6*sqrt(-a**2*x** 2 + 1) + 6*a**2*x**5*sqrt(-a**2*x**2 + 1) - 4*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 4*a**3*x**6*sqrt(-a**2*x**2 + 1) + 6*a**2*x**5*sqrt(-a**2*x**2 + 1) - 4*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**4
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (166) = 332\).
Time = 0.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.04 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=-\frac {29 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c^{4} {\left | a \right |}} - \frac {{\left (105 \, a^{3} + \frac {1365 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {51167 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac {260729 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac {621537 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac {826175 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac {642005 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}} + \frac {274995 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{11} x^{7}} - \frac {52500 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{8}}{a^{13} x^{8}}\right )} a^{4} x^{2}}{840 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} - \frac {\frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{4} {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4} {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{8}} \]
-29/2*a^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c ^4*abs(a)) - 1/840*(105*a^3 + 1365*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 5 1167*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a*x^2) + 260729*(sqrt(-a^2*x^2 + 1 )*abs(a) + a)^3/(a^3*x^3) - 621537*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^5* x^4) + 826175*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^7*x^5) - 642005*(sqrt(- a^2*x^2 + 1)*abs(a) + a)^6/(a^9*x^6) + 274995*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^11*x^7) - 52500*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^8/(a^13*x^8))*a^4 *x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a)) - 1/8*(20*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*c^4* abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4*abs(a)/(a*x^2))/(a^2*c^8)
Time = 3.43 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^4} \, dx=\frac {11\,a^4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^6\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^4\,x^2}+\frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {5\,a\,\sqrt {1-a^2\,x^2}}{c^4\,x}+\frac {1867\,a^3\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {41\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,29{}\mathrm {i}}{2\,c^4} \]
(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*29i)/(2*c^4) + (11*a^4*(1 - a^2*x^2)^(1/ 2))/(3*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) + (4*a^6*(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)/(2*c^4* x^2) + (2*a^4*(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)) - (5*a*(1 - a^2*x^2)^(1/2))/(c^4*x) + ( 1867*a^3*(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)) + (41*a^3*(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)))