Integrand size = 23, antiderivative size = 115 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}-\frac {15}{8} a^4 c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
-15/8*a^4*c*arctanh((-a^2*x^2+1)^(1/2))-1/4*c*(-a^2*x^2+1)^(1/2)/x^4-a*c*( -a^2*x^2+1)^(1/2)/x^3-15/8*a^2*c*(-a^2*x^2+1)^(1/2)/x^2-3*a^3*c*(-a^2*x^2+ 1)^(1/2)/x
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=\frac {1}{2} a c \left (-\frac {\sqrt {1-a^2 x^2} \left (2+3 a x+6 a^2 x^2\right )}{x^3}-3 a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-2 a^3 \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-a^2 x^2\right )\right ) \]
(a*c*(-((Sqrt[1 - a^2*x^2]*(2 + 3*a*x + 6*a^2*x^2))/x^3) - 3*a^3*ArcTanh[S qrt[1 - a^2*x^2]] - 2*a^3*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - a^2*x^2]))/2
Time = 0.43 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6698, 540, 25, 2338, 27, 539, 25, 27, 534, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle c \int \frac {(a x+1)^3}{x^5 \sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle c \left (-\frac {1}{4} \int -\frac {4 x^2 a^3+15 x a^2+12 a}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{4} \int \frac {4 x^2 a^3+15 x a^2+12 a}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle c \left (\frac {1}{4} \left (-\frac {1}{3} \int -\frac {9 a^2 (4 a x+5)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \int \frac {4 a x+5}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (-\frac {1}{2} \int -\frac {a (5 a x+8)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} \int \frac {a (5 a x+8)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} a \int \frac {5 a x+8}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} a \left (5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} a \left (\frac {5}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} a \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (\frac {1}{4} \left (3 a^2 \left (\frac {1}{2} a \left (-5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )\) |
c*(-1/4*Sqrt[1 - a^2*x^2]/x^4 + ((-4*a*Sqrt[1 - a^2*x^2])/x^3 + 3*a^2*((-5 *Sqrt[1 - a^2*x^2])/(2*x^2) + (a*((-8*Sqrt[1 - a^2*x^2])/x - 5*a*ArcTanh[S qrt[1 - a^2*x^2]]))/2))/4)
3.12.45.3.1 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[(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[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*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*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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]
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {\left (24 a^{5} x^{5}+15 a^{4} x^{4}-16 a^{3} x^{3}-13 a^{2} x^{2}-8 a x -2\right ) c}{8 x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{8}\) | \(77\) |
| default | \(-c \left (\frac {a^{5} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {13 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+3 a^{4} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-3 a \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )+2 a^{3} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(231\) |
| meijerg | \(-\frac {2 a^{4} c \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{\sqrt {\pi }}+\frac {a^{4} c \left (\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}\right )}{\sqrt {\pi }}-\frac {a^{5} c x}{\sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{3} c \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}-\frac {3 a^{4} c \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {a c \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{x^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(425\) |
1/8*(24*a^5*x^5+15*a^4*x^4-16*a^3*x^3-13*a^2*x^2-8*a*x-2)/x^4/(-a^2*x^2+1) ^(1/2)*c-15/8*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))*c
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=\frac {15 \, a^{4} c x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (24 \, a^{3} c x^{3} + 15 \, a^{2} c x^{2} + 8 \, a c x + 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, x^{4}} \]
1/8*(15*a^4*c*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (24*a^3*c*x^3 + 15*a^2 *c*x^2 + 8*a*c*x + 2*c)*sqrt(-a^2*x^2 + 1))/x^4
Result contains complex when optimal does not.
Time = 7.62 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.57 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=a^{3} c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
a**3*c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a **2*x**2 + 1)/x, True)) + 3*a**2*c*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/A bs(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*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*s qrt(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*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))
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.30 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=\frac {3 \, a^{5} c x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {15}{8} \, a^{4} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {15 \, a^{4} c}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {2 \, a^{3} c}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {13 \, a^{2} c}{8 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {a c}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {c}{4 \, \sqrt {-a^{2} x^{2} + 1} x^{4}} \]
3*a^5*c*x/sqrt(-a^2*x^2 + 1) - 15/8*a^4*c*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 15/8*a^4*c/sqrt(-a^2*x^2 + 1) - 2*a^3*c/(sqrt(-a^2*x^2 + 1)* x) - 13/8*a^2*c/(sqrt(-a^2*x^2 + 1)*x^2) - a*c/(sqrt(-a^2*x^2 + 1)*x^3) - 1/4*c/(sqrt(-a^2*x^2 + 1)*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (99) = 198\).
Time = 0.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.43 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=\frac {{\left (a^{5} c + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c}{x^{2}} + \frac {104 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{a x^{3}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {15 \, a^{5} c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {104 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c {\left | a \right |}}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c {\left | a \right |}}{x^{3}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \]
1/64*(a^5*c + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3*c/x + 32*(sqrt(-a^2*x^ 2 + 1)*abs(a) + a)^2*a*c/x^2 + 104*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c/(a* x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)) - 15/8*a^5*c*log( 1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/64*(1 04*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*c*abs(a)/x + 32*(sqrt(-a^2*x^2 + 1) *abs(a) + a)^2*a^3*c*abs(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c* abs(a)/x^3 + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c*abs(a)/(a*x^4))/a^4
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^5} \, dx=-\frac {c\,\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {15\,a^2\,c\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {3\,a^3\,c\,\sqrt {1-a^2\,x^2}}{x}-\frac {a\,c\,\sqrt {1-a^2\,x^2}}{x^3}+\frac {a^4\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8} \]