Integrand size = 19, antiderivative size = 110 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=-\frac {c^4 \sqrt {1-a^2 x^2}}{4 x^4}+\frac {a c^4 \sqrt {1-a^2 x^2}}{x^3}-\frac {11 a^2 c^4 \sqrt {1-a^2 x^2}}{8 x^2}+a^4 c^4 \arcsin (a x)+\frac {13}{8} a^4 c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
a^4*c^4*arcsin(a*x)+13/8*a^4*c^4*arctanh((-a^2*x^2+1)^(1/2))-1/4*c^4*(-a^2 *x^2+1)^(1/2)/x^4+a*c^4*(-a^2*x^2+1)^(1/2)/x^3-11/8*a^2*c^4*(-a^2*x^2+1)^( 1/2)/x^2
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=\frac {1}{16} c^4 \left (-13 a^4 \arcsin (a x)-\frac {2 \left (2-8 a x+9 a^2 x^2+8 a^3 x^3-11 a^4 x^4+29 a^4 x^4 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{x^4 \sqrt {1-a^2 x^2}}+26 a^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right ) \]
(c^4*(-13*a^4*ArcSin[a*x] - (2*(2 - 8*a*x + 9*a^2*x^2 + 8*a^3*x^3 - 11*a^4 *x^4 + 29*a^4*x^4*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(x^4*S qrt[1 - a^2*x^2]) + 26*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]))/16
Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6678, 27, 540, 2338, 27, 537, 25, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{x^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int \frac {(1-a x)^3 \sqrt {1-a^2 x^2}}{x^5}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle c^4 \left (-\frac {1}{4} \int \frac {\sqrt {1-a^2 x^2} \left (4 x^2 a^3-13 x a^2+12 a\right )}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {3 a^2 (13-4 a x) \sqrt {1-a^2 x^2}}{x^3}dx+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \int \frac {(13-4 a x) \sqrt {1-a^2 x^2}}{x^3}dx+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (\frac {1}{2} a^2 \int -\frac {13-8 a x}{x \sqrt {1-a^2 x^2}}dx-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \int \frac {13-8 a x}{x \sqrt {1-a^2 x^2}}dx-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \left (13 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-8 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \left (13 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-8 \arcsin (a x)\right )-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \left (\frac {13}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-8 \arcsin (a x)\right )-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \left (-\frac {13 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-8 \arcsin (a x)\right )-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c^4 \left (\frac {1}{4} \left (a^2 \left (-\frac {1}{2} a^2 \left (-13 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-8 \arcsin (a x)\right )-\frac {(13-8 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {4 a \left (1-a^2 x^2\right )^{3/2}}{x^3}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{4 x^4}\right )\) |
c^4*(-1/4*(1 - a^2*x^2)^(3/2)/x^4 + ((4*a*(1 - a^2*x^2)^(3/2))/x^3 + a^2*( -1/2*((13 - 8*a*x)*Sqrt[1 - a^2*x^2])/x^2 - (a^2*(-8*ArcSin[a*x] - 13*ArcT anh[Sqrt[1 - a^2*x^2]]))/2))/4)
3.4.24.3.1 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[(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_.))*((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]
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(\frac {\left (11 a^{4} x^{4}-8 a^{3} x^{3}-9 a^{2} x^{2}+8 a x -2\right ) c^{4}}{8 x^{4} \sqrt {-a^{2} x^{2}+1}}+\left (\frac {a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {13 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}\right ) c^{4}\) | \(104\) |
| default | \(c^{4} \left (\frac {a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {11 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}+3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {2 a^{3} \sqrt {-a^{2} x^{2}+1}}{x}\right )\) | \(172\) |
| meijerg | \(a^{4} c^{4} \arcsin \left (a x \right )-\frac {3 a^{4} c^{4} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}-\frac {2 a^{3} c^{4} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a^{4} c^{4} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{\sqrt {\pi }}+\frac {a \,c^{4} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{x^{3}}+\frac {a^{4} c^{4} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}\) | \(365\) |
1/8*(11*a^4*x^4-8*a^3*x^3-9*a^2*x^2+8*a*x-2)/x^4/(-a^2*x^2+1)^(1/2)*c^4+(a ^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+13/8*a^4*arctanh(1 /(-a^2*x^2+1)^(1/2)))*c^4
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=-\frac {16 \, a^{4} c^{4} x^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 13 \, a^{4} c^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (11 \, a^{2} c^{4} x^{2} - 8 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, x^{4}} \]
-1/8*(16*a^4*c^4*x^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 13*a^4*c^4*x ^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (11*a^2*c^4*x^2 - 8*a*c^4*x + 2*c^4)* sqrt(-a^2*x^2 + 1))/x^4
Result contains complex when optimal does not.
Time = 5.51 (sec) , antiderivative size = 502, normalized size of antiderivative = 4.56 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=a^{5} c^{4} \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 ) - 3 a^{4} c^{4} \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 ) + 2 a^{3} c^{4} \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 ) + 2 a^{2} c^{4} \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 ) - 3 a c^{4} \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 ) + c^{4} \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*c**4*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/s qrt(-a**2), Ne(a**2, 0)), (x, True)) - 3*a**4*c**4*Piecewise((-acosh(1/(a* x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**3*c**4*Piecewi se((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/ x, True)) + 2*a**2*c**4*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 )) - 3*a*c**4*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)) + c**4*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))
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=a^{4} c^{4} \arcsin \left (a x\right ) + \frac {13}{8} \, a^{4} c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{4}}{x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{4 \, x^{4}} \]
a^4*c^4*arcsin(a*x) + 13/8*a^4*c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs (x)) - 11/8*sqrt(-a^2*x^2 + 1)*a^2*c^4/x^2 + sqrt(-a^2*x^2 + 1)*a*c^4/x^3 - 1/4*sqrt(-a^2*x^2 + 1)*c^4/x^4
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (96) = 192\).
Time = 0.29 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.87 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=\frac {a^{5} c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {13 \, a^{5} c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {{\left (a^{5} c^{4} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{4}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c^{4}}{x^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a x^{3}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{4} {\left | a \right |}}{x} - \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c^{4} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{4} {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \]
a^5*c^4*arcsin(a*x)*sgn(a)/abs(a) + 13/8*a^5*c^4*log(1/2*abs(-2*sqrt(-a^2* x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/64*(a^5*c^4 - 8*(sqrt(-a^2 *x^2 + 1)*abs(a) + a)*a^3*c^4/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a*c ^4/x^2 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a*x^3))*a^8*x^4/((sqrt(- a^2*x^2 + 1)*abs(a) + a)^4*abs(a)) + 1/64*(8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*c^4*abs(a)/x - 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*c^4*abs(a)/ x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c^4*abs(a)/x^3 - (sqrt(-a^2*x^ 2 + 1)*abs(a) + a)^4*c^4*abs(a)/(a*x^4))/a^4
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x^5} \, dx=\frac {a\,c^4\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {11\,a^2\,c^4\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {a^4\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{8} \]