Integrand size = 22, antiderivative size = 157 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \arcsin (a x)}{a}-\frac {3 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \]
1/8*c^3*(a*x+8)*(-a^2*x^2+1)^(3/2)/a^4/x^3+1/5*c^3*(-a^2*x^2+1)^(5/2)/a^6/ x^5+3/4*c^3*(-a^2*x^2+1)^(5/2)/a^5/x^4-3*c^3*arcsin(a*x)/a-3/8*c^3*arctanh ((-a^2*x^2+1)^(1/2))/a-3/8*c^3*(-a*x+8)*(-a^2*x^2+1)^(1/2)/a^2/x
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.18 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (8+30 a x-24 a^2 x^2-105 a^3 x^3+24 a^4 x^4+75 a^5 x^5-8 a^6 x^6+45 a^5 x^5 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+40 a^2 x^2 \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},a^2 x^2\right )-8 a^5 x^5 \left (-1+a^2 x^2\right )^3 \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1-a^2 x^2\right )\right )}{40 a^6 x^5 \sqrt {1-a^2 x^2}} \]
(c^3*(8 + 30*a*x - 24*a^2*x^2 - 105*a^3*x^3 + 24*a^4*x^4 + 75*a^5*x^5 - 8* a^6*x^6 + 45*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]] + 40*a^2 *x^2*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, a^2*x^2] - 8*a^ 5*x^5*(-1 + a^2*x^2)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 - a^2*x^2]))/(40*a ^6*x^5*Sqrt[1 - a^2*x^2])
Time = 0.59 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6707, 6698, 540, 27, 2338, 25, 27, 537, 27, 536, 538, 223, 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.
| \(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle -\frac {c^3 \int \frac {e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle -\frac {c^3 \int \frac {(a x+1)^3 \left (1-a^2 x^2\right )^{3/2}}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{5} \int -\frac {5 \left (1-a^2 x^2\right )^{3/2} \left (x^2 a^3+3 x a^2+3 a\right )}{x^5}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (\int \frac {\left (1-a^2 x^2\right )^{3/2} \left (x^2 a^3+3 x a^2+3 a\right )}{x^5}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} \int -\frac {a^2 (a x+12) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} \int \frac {a^2 (a x+12) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \int \frac {(a x+12) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (a x+8) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(a x+8) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {a-8 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(8-a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-8 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(8-a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (8-a x)}{x}-8 a \arcsin (a x)\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (8-a x)}{x}-8 a \arcsin (a x)\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (8-a x)}{x}-8 a \arcsin (a x)\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (8-a x)}{x}-8 a \arcsin (a x)\right )-\frac {(a x+8) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}-\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
-((c^3*(-1/5*(1 - a^2*x^2)^(5/2)/x^5 - (3*a*(1 - a^2*x^2)^(5/2))/(4*x^4) + (a^2*(-1/2*((8 + a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (3*a^2*(-(((8 - a*x)*Sqr t[1 - a^2*x^2])/x) - 8*a*ArcSin[a*x] - a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/ 4))/a^6)
3.7.46.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Time = 0.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {\left (152 a^{6} x^{6}+55 a^{5} x^{5}-176 a^{4} x^{4}-85 a^{3} x^{3}+16 a^{2} x^{2}+30 a x +8\right ) c^{3}}{40 x^{5} \sqrt {-a^{2} x^{2}+1}\, a^{6}}-\frac {\left (\frac {3 a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-a^{5} \sqrt {-a^{2} x^{2}+1}\right ) c^{3}}{a^{6}}\) | \(145\) |
| default | \(\frac {c^{3} \left (a^{9} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {8 a^{6} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {1}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {6 a^{2} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )}{5}+3 a^{8} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-6 a^{5} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-3 a \left (-\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )+8 a^{3} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+6 a^{4} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a^{6}}\) | \(410\) |
| meijerg | \(-\frac {8 c^{3} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {6 c^{3} \left (-2 a^{2} x^{2}+1\right )}{a^{2} x \sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (-16 a^{6} x^{6}+8 a^{4} x^{4}+2 a^{2} x^{2}+1\right )}{5 a^{6} x^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {6 c^{3} \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }\, a}-\frac {8 c^{3} \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}\right )}{a \sqrt {\pi }}\) | \(550\) |
1/40*(152*a^6*x^6+55*a^5*x^5-176*a^4*x^4-85*a^3*x^3+16*a^2*x^2+30*a*x+8)/x ^5/(-a^2*x^2+1)^(1/2)*c^3/a^6-(3*a^6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^ 2*x^2+1)^(1/2))+3/8*a^5*arctanh(1/(-a^2*x^2+1)^(1/2))-a^5*(-a^2*x^2+1)^(1/ 2))*c^3/a^6
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 15 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 40 \, a^{5} c^{3} x^{5} + {\left (40 \, a^{5} c^{3} x^{5} - 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} + 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x + 8 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}} \]
1/40*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 15*a^5*c^3* x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 40*a^5*c^3*x^5 + (40*a^5*c^3*x^5 - 1 52*a^4*c^3*x^4 - 55*a^3*c^3*x^3 + 24*a^2*c^3*x^2 + 30*a*c^3*x + 8*c^3)*sqr t(-a^2*x^2 + 1))/(a^6*x^5)
Time = 10.23 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.35 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=- a c^{3} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 3 c^{3} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - \frac {c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {5 c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {5 c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {c^{3} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} - \frac {3 c^{3} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{5}} - \frac {c^{3} \left (\begin {cases} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {otherwise} \end {cases}\right )}{a^{6}} \]
-a*c**3*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True )) - 3*c**3*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1)) /sqrt(-a**2), Ne(a**2, 0)), (x, True)) - c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 5*c**3*Piecewise((-I*s qrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True) )/a**2 + 5*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a* *2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 - c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - s qrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 3*c**3*Piecewise((-3*a**4*acos h(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2 ))), True))/a**5 - c**3*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4 *a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x **4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a **3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x** 4), True))/a**6
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.82 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-a^{3} c^{3} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{3} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {8 \, c^{3} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {6 \, c^{3} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {6 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{3}}{a^{2}} - \frac {4 \, {\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{a^{3}} + \frac {3 \, {\left (15 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {15 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\right )} c^{3}}{8 \, a^{5}} - \frac {{\left (\frac {16 \, a^{6} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {8 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {2 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{5}}\right )} c^{3}}{5 \, a^{6}} \]
-a^3*c^3*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a ^2*c^3*(x/(sqrt(-a^2*x^2 + 1)*a^2) - arcsin(a*x)/a^3) - 8*c^3*x/sqrt(-a^2* x^2 + 1) - 6*c^3*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)))/a + 6*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))* c^3/a^2 - 4*(3*a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 3*a^2/sqr t(-a^2*x^2 + 1) + 1/(sqrt(-a^2*x^2 + 1)*x^2))*c^3/a^3 + 3/8*(15*a^4*log(2* sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 15*a^4/sqrt(-a^2*x^2 + 1) + 5*a^2/ (sqrt(-a^2*x^2 + 1)*x^2) + 2/(sqrt(-a^2*x^2 + 1)*x^4))*c^3/a^5 - 1/5*(16*a ^6*x/sqrt(-a^2*x^2 + 1) - 8*a^4/(sqrt(-a^2*x^2 + 1)*x) - 2*a^2/(sqrt(-a^2* x^2 + 1)*x^3) - 1/(sqrt(-a^2*x^2 + 1)*x^5))*c^3/a^6
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.45 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {{\left (2 \, c^{3} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} - \frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{320 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {3 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} - \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{320 \, a^{4} {\left | a \right |}} \]
-1/320*(2*c^3 + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) + 30*(sqrt( -a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4*x^2) - 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) - 580*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^8*x^4) )*a^10*x^5/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) - 3*c^3*arcsin(a*x)* sgn(a)/abs(a) - 3/8*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a ^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c^3/a - 1/320*(580*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x + 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 30*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^2*x^3) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^4*x^4) - 2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^3/ (a^6*x^5))/(a^4*abs(a))
Time = 0.07 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {19\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^2\,x}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^4\,x^3}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}+\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a} \]
(c^3*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/(8*a) - (3*c^3*asinh(x*(-a^2)^(1/2)) )/(-a^2)^(1/2) + (c^3*(1 - a^2*x^2)^(1/2))/a - (19*c^3*(1 - a^2*x^2)^(1/2) )/(5*a^2*x) - (11*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) + (3*c^3*(1 - a^2*x ^2)^(1/2))/(5*a^4*x^3) + (3*c^3*(1 - a^2*x^2)^(1/2))/(4*a^5*x^4) + (c^3*(1 - a^2*x^2)^(1/2))/(5*a^6*x^5)