Integrand size = 22, antiderivative size = 169 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \arcsin (a x)}{a}-\frac {35 c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{16 a} \] Output:
1/16*c^4*(35*a*x+16)*(-a^2*x^2+1)^(1/2)/a^2/x-1/48*c^4*(-35*a*x+16)*(-a^2* x^2+1)^(3/2)/a^4/x^3+1/120*c^4*(-35*a*x+24)*(-a^2*x^2+1)^(5/2)/a^6/x^5-1/4 2*c^4*(-7*a*x+6)*(-a^2*x^2+1)^(7/2)/a^8/x^7+c^4*arcsin(a*x)/a-35/16*c^4*ar ctanh((-a^2*x^2+1)^(1/2))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.41 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (-\frac {9 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {7}{2},-\frac {5}{2},a^2 x^2\right )}{x^7}+7 a^7 \left (1-a^2 x^2\right )^{9/2} \operatorname {Hypergeometric2F1}\left (4,\frac {9}{2},\frac {11}{2},1-a^2 x^2\right )\right )}{63 a^8} \] Input:
Integrate[(c - c/(a^2*x^2))^4/E^ArcTanh[a*x],x]
Output:
(c^4*((-9*Hypergeometric2F1[-7/2, -7/2, -5/2, a^2*x^2])/x^7 + 7*a^7*(1 - a ^2*x^2)^(9/2)*Hypergeometric2F1[4, 9/2, 11/2, 1 - a^2*x^2]))/(63*a^8)
Time = 0.85 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6707, 6699, 537, 25, 537, 25, 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.
| \(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle \frac {c^4 \int \frac {e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )^4}{x^8}dx}{a^8}\) |
| \(\Big \downarrow \) 6699 |
| \(\displaystyle \frac {c^4 \int \frac {(1-a x) \left (1-a^2 x^2\right )^{7/2}}{x^8}dx}{a^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{6} a^2 \int -\frac {(6-7 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \int \frac {(6-7 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (\frac {1}{4} a^2 \int -\frac {(24-35 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \int \frac {(24-35 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (16-35 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(16-35 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {-16 x a^2-35 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(35 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-16 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-35 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(35 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-35 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (35 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {35}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (35 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {35 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (35 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (35 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (35 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )}{a^8}\) |
Input:
Int[(c - c/(a^2*x^2))^4/E^ArcTanh[a*x],x]
Output:
(c^4*(-1/42*((6 - 7*a*x)*(1 - a^2*x^2)^(7/2))/x^7 - (a^2*(-1/20*((24 - 35* a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/2*((16 - 35*a*x)*(1 - a^2*x^2)^(3 /2))/x^3 - (3*a^2*(-(((16 + 35*a*x)*Sqrt[1 - a^2*x^2])/x) - 16*a*ArcSin[a* x] + 35*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/6))/a^8
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[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]) && ILtQ[(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 = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {\left (2816 a^{8} x^{8}+3045 a^{7} x^{7}-4768 x^{6} a^{6}-4375 a^{5} x^{5}+3008 a^{4} x^{4}+1610 a^{3} x^{3}-1296 a^{2} x^{2}-280 a x +240\right ) c^{4}}{1680 x^{7} \sqrt {-a^{2} x^{2}+1}\, a^{8}}-\frac {\left (\frac {35 a^{7} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{8} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{7} \sqrt {-a^{2} x^{2}+1}\right ) c^{4}}{a^{8}}\) | \(161\) |
| default | \(\frac {c^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 x^{7}}-\frac {17 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 x^{5}}-\frac {2 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 x^{3}}\right )}{7}+a^{7} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 x^{6}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )}{2}\right )+3 a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )-\frac {a^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{3}}-3 a^{5} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-a^{6} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{8}}\) | \(427\) |
Input:
int((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/1680*(2816*a^8*x^8+3045*a^7*x^7-4768*a^6*x^6-4375*a^5*x^5+3008*a^4*x^4+ 1610*a^3*x^3-1296*a^2*x^2-280*a*x+240)/x^7/(-a^2*x^2+1)^(1/2)*c^4/a^8-(35/ 16*a^7*arctanh(1/(-a^2*x^2+1)^(1/2))-a^8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/ (-a^2*x^2+1)^(1/2))-a^7*(-a^2*x^2+1)^(1/2))*c^4/a^8
Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3675 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 1680 \, a^{7} c^{4} x^{7} - {\left (1680 \, a^{7} c^{4} x^{7} + 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} - 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} + 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \] Input:
integrate((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas" )
Output:
-1/1680*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3675*a^ 7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1680*a^7*c^4*x^7 - (1680*a^7*c ^4*x^7 + 2816*a^6*c^4*x^6 + 3045*a^5*c^4*x^5 - 1952*a^4*c^4*x^4 - 1330*a^3 *c^4*x^3 + 1056*a^2*c^4*x^2 + 280*a*c^4*x - 240*c^4)*sqrt(-a^2*x^2 + 1))/( a^8*x^7)
Result contains complex when optimal does not.
Time = 13.02 (sec) , antiderivative size = 1110, normalized size of antiderivative = 6.57 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\text {Too large to display} \] Input:
integrate((c-c/a**2/x**2)**4/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
Output:
c**4*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*as in(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**4*Piecewise((-I*a**2*x/sqr t(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x* *2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x** 2 + 1)), True))/a**2 - 3*c**4*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sq rt(-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**4*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt (-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/( a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + 3*c**4 *Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*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), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*s qrt(1 - 1/(a**2*x**2))), True))/a**5 - 3*c**4*Piecewise((2*I*a**4*sqrt(a** 2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x* *2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), Tr ue))/a**6 - c**4*Piecewise((a**6*acosh(1/(a*x))/16 - a**5/(16*x*sqrt(-1...
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}{a x + 1} \,d x } \] Input:
integrate((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima" )
Output:
c^4*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - integrate((4*a^6*c^4*x^6 - 6* a^4*c^4*x^4 + 4*a^2*c^4*x^2 - c^4)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^9*x^9 + a^8*x^8), x)
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (149) = 298\).
Time = 0.14 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.98 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {{\left (15 \, c^{4} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} - \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} + \frac {c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {35 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {\frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6} {\left | a \right |}} \] Input:
integrate((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
1/13440*(15*c^4 - 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) - 189*(sq rt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4*x^2) + 525*(sqrt(-a^2*x^2 + 1)*abs (a) + a)^3*c^4/(a^6*x^3) + 1295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^8 *x^4) - 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) - 9765*(sqrt (-a^2*x^2 + 1)*abs(a) + a)^6*c^4/(a^12*x^6))*a^14*x^7/((sqrt(-a^2*x^2 + 1) *abs(a) + a)^7*abs(a)) + c^4*arcsin(a*x)*sgn(a)/abs(a) - 35/16*c^4*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^4/a + 1/13440*(9765*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x + 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 1295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/x^3 - 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^ 2*x^4) + 189*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^4*x^5) + 35*(sqrt(-a ^2*x^2 + 1)*abs(a) + a)^6*c^4/(a^6*x^6) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7*c^4/(a^8*x^7))/(a^6*abs(a))
Time = 22.76 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.34 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}+\frac {176\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^2\,x}+\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}-\frac {122\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4\,x^3}-\frac {19\,c^4\,\sqrt {1-a^2\,x^2}}{24\,a^5\,x^4}+\frac {22\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{6\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,35{}\mathrm {i}}{16\,a} \] Input:
int(((c - c/(a^2*x^2))^4*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)
Output:
(c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^4*atan((1 - a^2*x^2)^(1/2)*1 i)*35i)/(16*a) + (c^4*(1 - a^2*x^2)^(1/2))/a + (176*c^4*(1 - a^2*x^2)^(1/2 ))/(105*a^2*x) + (29*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) - (122*c^4*(1 - a^2*x^2)^(1/2))/(105*a^4*x^3) - (19*c^4*(1 - a^2*x^2)^(1/2))/(24*a^5*x^4) + (22*c^4*(1 - a^2*x^2)^(1/2))/(35*a^6*x^5) + (c^4*(1 - a^2*x^2)^(1/2))/( 6*a^7*x^6) - (c^4*(1 - a^2*x^2)^(1/2))/(7*a^8*x^7)
Time = 0.15 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.12 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (13440 \mathit {asin} \left (a x \right ) a^{7} x^{7}+13440 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+22528 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+24360 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-15616 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-10640 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8448 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2240 \sqrt {-a^{2} x^{2}+1}\, a x -1920 \sqrt {-a^{2} x^{2}+1}+29400 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{7} x^{7}-18375 a^{7} x^{7}\right )}{13440 a^{8} x^{7}} \] Input:
int((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
Output:
(c**4*(13440*asin(a*x)*a**7*x**7 + 13440*sqrt( - a**2*x**2 + 1)*a**7*x**7 + 22528*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 24360*sqrt( - a**2*x**2 + 1)*a* *5*x**5 - 15616*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 10640*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 8448*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 2240*sqrt( - a** 2*x**2 + 1)*a*x - 1920*sqrt( - a**2*x**2 + 1) + 29400*log(tan(asin(a*x)/2) )*a**7*x**7 - 18375*a**7*x**7))/(13440*a**8*x**7)