Integrand size = 20, antiderivative size = 136 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 (8-15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \arcsin (a x)}{a}+\frac {15 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \] Output:
1/8*c^3*(-15*a*x+8)*(-a^2*x^2+1)^(1/2)/a^2/x-1/24*c^3*(15*a*x+8)*(-a^2*x^2 +1)^(3/2)/a^4/x^3+1/20*c^3*(5*a*x+4)*(-a^2*x^2+1)^(5/2)/a^6/x^5+c^3*arcsin (a*x)/a+15/8*c^3*arctanh((-a^2*x^2+1)^(1/2))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.51 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\frac {7 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},a^2 x^2\right )}{x^5}+5 a^5 \left (1-a^2 x^2\right )^{7/2} \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},1-a^2 x^2\right )\right )}{35 a^6} \] Input:
Integrate[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^3,x]
Output:
(c^3*((7*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2])/x^5 + 5*a^5*(1 - a^ 2*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 - a^2*x^2]))/(35*a^6)
Time = 0.49 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6707, 6698, 537, 25, 537, 25, 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.
| \(\displaystyle \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle -\frac {c^3 \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle -\frac {c^3 \int \frac {(a x+1) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \int -\frac {(5 a x+4) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \int \frac {(5 a x+4) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {(15 a x+8) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \int \frac {(15 a x+8) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (\int \frac {15 a-8 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(8-15 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-8 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+15 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(8-15 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (15 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (8-15 a x)}{x}-8 a \arcsin (a x)\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (\frac {15}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (8-15 a x)}{x}-8 a \arcsin (a x)\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-\frac {15 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (8-15 a x)}{x}-8 a \arcsin (a x)\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{4} a^2 \left (-\frac {1}{2} a^2 \left (-15 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (8-15 a x)}{x}-8 a \arcsin (a x)\right )-\frac {(15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {(5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )}{a^6}\) |
Input:
Int[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^3,x]
Output:
-((c^3*(-1/20*((4 + 5*a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/6*((8 + 15* a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (a^2*(-(((8 - 15*a*x)*Sqrt[1 - a^2*x^2])/x ) - 8*a*ArcSin[a*x] - 15*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/a^6)
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[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 = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(-\frac {\left (184 x^{6} a^{6}-135 a^{5} x^{5}-272 a^{4} x^{4}+165 a^{3} x^{3}+112 a^{2} x^{2}-30 a x -24\right ) c^{3}}{120 x^{5} \sqrt {-a^{2} x^{2}+1}\, a^{6}}-\frac {\left (-\frac {15 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-\frac {a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a^{5} \sqrt {-a^{2} x^{2}+1}\right ) c^{3}}{a^{6}}\) | \(144\) |
| default | \(\frac {c^{3} \left (\frac {a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{5} \sqrt {-a^{2} x^{2}+1}+\frac {\sqrt {-a^{2} x^{2}+1}}{5 x^{5}}+\frac {11 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )}{5}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )+3 a^{3} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+\frac {3 a^{4} \sqrt {-a^{2} x^{2}+1}}{x}+3 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a^{6}}\) | \(256\) |
| meijerg | \(-\frac {c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 a \sqrt {\pi }}-\frac {c^{3} \left (-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}\right )}{2 a \sqrt {\pi }}+\frac {c^{3} \arcsin \left (a x \right )}{a}+\frac {3 c^{3} \sqrt {-a^{2} x^{2}+1}}{a^{2} x}-\frac {c^{3} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{a^{4} x^{3}}+\frac {c^{3} \left (\frac {8}{3} a^{4} x^{4}+\frac {4}{3} a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{5 a^{6} x^{5}}\) | \(443\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/120*(184*a^6*x^6-135*a^5*x^5-272*a^4*x^4+165*a^3*x^3+112*a^2*x^2-30*a*x -24)/x^5/(-a^2*x^2+1)^(1/2)*c^3/a^6-(-15/8*a^5*arctanh(1/(-a^2*x^2+1)^(1/2 ))-a^6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+a^5*(-a^2*x^2+ 1)^(1/2))*c^3/a^6
Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 225 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 120 \, a^{5} c^{3} x^{5} + {\left (120 \, a^{5} c^{3} x^{5} - 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} + 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x - 24 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x, algorithm="fricas" )
Output:
-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 225*a^5*c ^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 120*a^5*c^3*x^5 + (120*a^5*c^3*x^ 5 - 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 + 88*a^2*c^3*x^2 - 30*a*c^3*x - 24*c ^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)
Time = 8.26 (sec) , antiderivative size = 683, normalized size of antiderivative = 5.02 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**3,x)
Output:
a*c**3*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True) ) + c**3*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/sq rt(-a**2), Ne(a**2, 0)), (x, True)) - 3*c**3*Piecewise((-acosh(1/(a*x)), 1 /Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 3*c**3*Piecewise((-I*sq rt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) /a**2 + 3*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 + 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) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - c**3*Piecewise((-3*a**4*acosh (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. 332 vs. \(2 (120) = 240\).
Time = 0.11 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.44 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \arcsin \left (a x\right )}{a} + \frac {3 \, c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {3 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{3}}{2 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x} - \frac {{\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{3}}{a^{4}} + \frac {{\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{x^{4}}\right )} c^{3}}{8 \, a^{5}} + \frac {{\left (\frac {8 \, \sqrt {-a^{2} x^{2} + 1} a^{4}}{x} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{x^{5}}\right )} c^{3}}{15 \, a^{6}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x, algorithm="maxima" )
Output:
c^3*arcsin(a*x)/a + 3*c^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a - sqrt(-a^2*x^2 + 1)*c^3/a - 3/2*(a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/ab s(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^3/a^3 + 3*sqrt(-a^2*x^2 + 1)*c^3/(a^2*x) - (2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(-a^2*x^2 + 1)/x^3)*c^3/a^4 + 1/8*(3* a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 3*sqrt(-a^2*x^2 + 1)*a^2 /x^2 + 2*sqrt(-a^2*x^2 + 1)/x^4)*c^3/a^5 + 1/15*(8*sqrt(-a^2*x^2 + 1)*a^4/ x + 4*sqrt(-a^2*x^2 + 1)*a^2/x^3 + 3*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. \(2 (120) = 240\).
Time = 0.14 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.83 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {{\left (6 \, c^{3} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} + \frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} + \frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {15 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} + \frac {\frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} - \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{960 \, a^{4} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x, algorithm="giac")
Output:
-1/960*(6*c^3 + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) - 70*(sqrt( -a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4*x^2) - 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) + 660*(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)) + c^3*arcsin(a*x)*s gn(a)/abs(a) + 15/8*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 + 1/960*(660*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x - 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 70*(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) + 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^3 /(a^6*x^5))/(a^4*abs(a))
Time = 14.47 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^2\,x}-\frac {9\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^4\,x^3}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8\,a} \] Input:
int(((c - c/(a^2*x^2))^3*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
Output:
(c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (c^3*atan((1 - a^2*x^2)^(1/2)*1 i)*15i)/(8*a) - (c^3*(1 - a^2*x^2)^(1/2))/a + (23*c^3*(1 - a^2*x^2)^(1/2)) /(15*a^2*x) - (9*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) - (11*c^3*(1 - a^2*x ^2)^(1/2))/(15*a^4*x^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)
Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (120 \mathit {asin} \left (a x \right ) a^{5} x^{5}-120 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+184 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-135 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-88 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+30 \sqrt {-a^{2} x^{2}+1}\, a x +24 \sqrt {-a^{2} x^{2}+1}-225 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{5} x^{5}+150 a^{5} x^{5}\right )}{120 a^{6} x^{5}} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x)
Output:
(c**3*(120*asin(a*x)*a**5*x**5 - 120*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 18 4*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 135*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 88*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 30*sqrt( - a**2*x**2 + 1)*a*x + 24 *sqrt( - a**2*x**2 + 1) - 225*log(tan(asin(a*x)/2))*a**5*x**5 + 150*a**5*x **5))/(120*a**6*x**5)