Integrand size = 19, antiderivative size = 137 \[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{7 c^4 (1-a x)^4}+\frac {13 \sqrt {1-a^2 x^2}}{35 c^4 (1-a x)^3}+\frac {61 \sqrt {1-a^2 x^2}}{105 c^4 (1-a x)^2}+\frac {166 \sqrt {1-a^2 x^2}}{105 c^4 (1-a x)}-\frac {\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \] Output:
2/7*(-a^2*x^2+1)^(1/2)/c^4/(-a*x+1)^4+13/35*(-a^2*x^2+1)^(1/2)/c^4/(-a*x+1 )^3+61/105*(-a^2*x^2+1)^(1/2)/c^4/(-a*x+1)^2+166/105*(-a^2*x^2+1)^(1/2)/c^ 4/(-a*x+1)-arctanh((-a^2*x^2+1)^(1/2))/c^4
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\frac {120+525 a x+105 a^2 x^2-700 a^3 x^3+581 a^5 x^5-166 a^7 x^7+15 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1-a^2 x^2\right )}{105 c^4 \left (1-a^2 x^2\right )^{7/2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x*(c - a*c*x)^4),x]
Output:
(120 + 525*a*x + 105*a^2*x^2 - 700*a^3*x^3 + 581*a^5*x^5 - 166*a^7*x^7 + 1 5*Hypergeometric2F1[-7/2, 1, -5/2, 1 - a^2*x^2])/(105*c^4*(1 - a^2*x^2)^(7 /2))
Time = 0.52 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {6678, 27, 570, 532, 25, 2336, 25, 532, 25, 532, 27, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^5 x (1-a x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x (1-a x)^5}dx}{c^4}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^5}{x \left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int -\frac {-7 a^3 x^3-35 a^2 x^2+19 a x+7}{x \left (1-a^2 x^2\right )^{7/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {-7 a^3 x^3-35 a^2 x^2+19 a x+7}{x \left (1-a^2 x^2\right )^{7/2}}dx+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{7} \left (-\frac {1}{5} \int -\frac {83 a x+35}{x \left (1-a^2 x^2\right )^{5/2}}dx-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {83 a x+35}{x \left (1-a^2 x^2\right )^{5/2}}dx-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {166 a x+105}{x \left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {166 a x+105}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {166 a x+105}{\sqrt {1-a^2 x^2}}-\int -\frac {105}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {166 a x+105}{\sqrt {1-a^2 x^2}}\right )+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {166 a x+105}{\sqrt {1-a^2 x^2}}\right )+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {166 a x+105}{\sqrt {1-a^2 x^2}}-\frac {105 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {166 a x+105}{\sqrt {1-a^2 x^2}}-105 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\frac {83 a x+35}{3 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {4 (7-3 a x)}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {16 (a x+1)}{7 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\) |
Input:
Int[E^ArcTanh[a*x]/(x*(c - a*c*x)^4),x]
Output:
((16*(1 + a*x))/(7*(1 - a^2*x^2)^(7/2)) + ((-4*(7 - 3*a*x))/(5*(1 - a^2*x^ 2)^(5/2)) + ((35 + 83*a*x)/(3*(1 - a^2*x^2)^(3/2)) + ((105 + 166*a*x)/Sqrt [1 - a^2*x^2] - 105*ArcTanh[Sqrt[1 - a^2*x^2]])/3)/5)/7)/c^4
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[((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. \(450\) vs. \(2(119)=238\).
Time = 0.29 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.29
| method | result | size |
| default | \(\frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{3}}-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}}{c^{4}}\) | \(451\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/c^4*(-arctanh(1/(-a^2*x^2+1)^(1/2))+1/a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2 -2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))+2/a^3* (1/7/a/(x-1/a)^4*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-3/7*a*(1/5/a/(x-1/a)^3 *(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2 -2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))))-1/a^ 2*(1/5/a/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a) ^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1 /a))^(1/2)))-1/a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))
Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\frac {296 \, a^{4} x^{4} - 1184 \, a^{3} x^{3} + 1776 \, a^{2} x^{2} - 1184 \, a x + 105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (166 \, a^{3} x^{3} - 559 \, a^{2} x^{2} + 659 \, a x - 296\right )} \sqrt {-a^{2} x^{2} + 1} + 296}{105 \, {\left (a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x, algorithm="fricas")
Output:
1/105*(296*a^4*x^4 - 1184*a^3*x^3 + 1776*a^2*x^2 - 1184*a*x + 105*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (1 66*a^3*x^3 - 559*a^2*x^2 + 659*a*x - 296)*sqrt(-a^2*x^2 + 1) + 296)/(a^4*c ^4*x^4 - 4*a^3*c^4*x^3 + 6*a^2*c^4*x^2 - 4*a*c^4*x + c^4)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\frac {\int \frac {a x}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a*c*x+c)**4,x)
Output:
(Integral(a*x/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 4*a**3*x**4*sqrt(-a**2*x** 2 + 1) + 6*a**2*x**3*sqrt(-a**2*x**2 + 1) - 4*a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 4*a**3*x**4*sqrt(-a**2*x**2 + 1) + 6*a**2*x**3*sqrt(-a**2*x**2 + 1) - 4 *a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**4
\[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x, algorithm="maxima")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(a*c*x - c)^4*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (115) = 230\).
Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.77 \[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=-\frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{4} {\left | a \right |}} + \frac {2 \, {\left (296 \, a - \frac {1547 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a x} + \frac {4011 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{3} x^{2}} - \frac {5600 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{5} x^{3}} + \frac {4760 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{7} x^{4}} - \frac {2205 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{9} x^{5}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{11} x^{6}}\right )}}{105 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x, algorithm="giac")
Output:
-a*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c^4*abs( a)) + 2/105*(296*a - 1547*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a*x) + 4011*(sq rt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^3*x^2) - 5600*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^5*x^3) + 4760*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^7*x^4) - 220 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^9*x^5) + 525*(sqrt(-a^2*x^2 + 1)*ab s(a) + a)^6/(a^11*x^6))/(c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1) ^7*abs(a))
Time = 14.16 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.39 \[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\frac {7\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {166\,a\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {13\,a\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^4} \] Input:
int((a*x + 1)/(x*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)
Output:
(atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^4 + (7*a^2*(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^4*(1 - a^2*x^2)^(1/2))/(35*(a^ 4*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) + (2*a^2*(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)) + (166*a *(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)) + (13*a*(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)))
\[ \int \frac {e^{\text {arctanh}(a x)}}{x (c-a c x)^4} \, dx=\int \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}\, x \left (-a c x +c \right )^{4}}d x \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x)
Output:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x)