Integrand size = 27, antiderivative size = 156 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=-\frac {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {3}{16} a^6 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
-1/6*(-a^2*c*x^2+c)^(3/2)/x^6-2/5*a*(-a^2*c*x^2+c)^(3/2)/x^5-3/8*a^2*(-a^2 *c*x^2+c)^(3/2)/x^4-4/15*a^3*(-a^2*c*x^2+c)^(3/2)/x^3+3/16*a^6*c^(3/2)*arc tanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))-3/16*a^4*c*(-a^2*c*x^2+c)^(1/2)/x^2
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=\frac {1}{240} c \left (\frac {\sqrt {c-a^2 c x^2} \left (-40-96 a x-50 a^2 x^2+32 a^3 x^3+45 a^4 x^4+64 a^5 x^5\right )}{x^6}-45 a^6 \sqrt {c} \log (x)+45 a^6 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )\right ) \]
(c*((Sqrt[c - a^2*c*x^2]*(-40 - 96*a*x - 50*a^2*x^2 + 32*a^3*x^3 + 45*a^4* x^4 + 64*a^5*x^5))/x^6 - 45*a^6*Sqrt[c]*Log[x] + 45*a^6*Sqrt[c]*Log[c + Sq rt[c]*Sqrt[c - a^2*c*x^2]]))/240
Time = 0.48 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6701, 540, 27, 539, 25, 27, 539, 25, 27, 534, 243, 51, 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.
| \(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 6701 |
| \(\displaystyle c \int \frac {(a x+1)^2 \sqrt {c-a^2 c x^2}}{x^7}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle c \left (-\frac {\int -\frac {3 a c (3 a x+4) \sqrt {c-a^2 c x^2}}{x^6}dx}{6 c}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} a \int \frac {(3 a x+4) \sqrt {c-a^2 c x^2}}{x^6}dx-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle c \left (\frac {1}{2} a \left (-\frac {\int -\frac {a c (8 a x+15) \sqrt {c-a^2 c x^2}}{x^5}dx}{5 c}-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {\int \frac {a c (8 a x+15) \sqrt {c-a^2 c x^2}}{x^5}dx}{5 c}-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \int \frac {(8 a x+15) \sqrt {c-a^2 c x^2}}{x^5}dx-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (-\frac {\int -\frac {a c (15 a x+32) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {\int \frac {a c (15 a x+32) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \int \frac {(15 a x+32) \sqrt {c-a^2 c x^2}}{x^4}dx-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (15 a \int \frac {\sqrt {c-a^2 c x^2}}{x^3}dx-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {15}{2} a \int \frac {\sqrt {c-a^2 c x^2}}{x^4}dx^2-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {15}{2} a \left (-\frac {1}{2} a^2 c \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {15}{2} a \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {15}{2} a \left (a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {15 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {4 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 c x^6}\right )\) |
c*(-1/6*(c - a^2*c*x^2)^(3/2)/(c*x^6) + (a*((-4*(c - a^2*c*x^2)^(3/2))/(5* c*x^5) + (a*((-15*(c - a^2*c*x^2)^(3/2))/(4*c*x^4) + (a*((-32*(c - a^2*c*x ^2)^(3/2))/(3*c*x^3) + (15*a*(-(Sqrt[c - a^2*c*x^2]/x^2) + a^2*Sqrt[c]*Arc Tanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]))/2))/4))/5))/2)
3.11.96.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*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/2, 0]
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(-\frac {\left (64 a^{7} x^{7}+45 a^{6} x^{6}-32 a^{5} x^{5}-95 a^{4} x^{4}-128 a^{3} x^{3}+10 a^{2} x^{2}+96 a x +40\right ) c^{2}}{240 x^{6} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 a^{6} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{16}\) | \(113\) |
| default | \(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}+\frac {13 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )}{4}\right )}{6}+2 a^{6} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )}{3}\right )+2 a^{4} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )+2 a^{5} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 c \,x^{5}}-2 a^{6} \left (\frac {\left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )\) | \(710\) |
-1/240*(64*a^7*x^7+45*a^6*x^6-32*a^5*x^5-95*a^4*x^4-128*a^3*x^3+10*a^2*x^2 +96*a*x+40)/x^6/(-c*(a^2*x^2-1))^(1/2)*c^2+3/16*a^6*c^(3/2)*ln((2*c+2*c^(1 /2)*(-a^2*c*x^2+c)^(1/2))/x)
Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.46 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=\left [\frac {45 \, a^{6} c^{\frac {3}{2}} x^{6} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, x^{6}}, \frac {45 \, a^{6} \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, x^{6}}\right ] \]
[1/480*(45*a^6*c^(3/2)*x^6*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c ) - 2*c)/x^2) + 2*(64*a^5*c*x^5 + 45*a^4*c*x^4 + 32*a^3*c*x^3 - 50*a^2*c*x ^2 - 96*a*c*x - 40*c)*sqrt(-a^2*c*x^2 + c))/x^6, 1/240*(45*a^6*sqrt(-c)*c* x^6*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (64*a^5*c*x^5 + 45*a^4*c*x^4 + 32*a^3*c*x^3 - 50*a^2*c*x^2 - 96*a*c*x - 40*c)*sqrt(-a^2* c*x^2 + c))/x^6]
Result contains complex when optimal does not.
Time = 9.58 (sec) , antiderivative size = 636, normalized size of antiderivative = 4.08 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=a^{2} c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {2 i a^{4} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{6} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} - \frac {a^{5} \sqrt {c}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {a^{3} \sqrt {c}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {5 a \sqrt {c}}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{6 a x^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{6} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} + \frac {i a^{5} \sqrt {c}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {i a^{3} \sqrt {c}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {5 i a \sqrt {c}}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
a**2*c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(- 1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt (c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*s qrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3* I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + 2*a*c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x* *2 - 1)/(15*x) + I*a**2*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(c)* sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a **2*x**2 + 1)/(15*x) + a**2*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt( c)*sqrt(-a**2*x**2 + 1)/(5*x**5), True)) + c*Piecewise((a**6*sqrt(c)*acosh (1/(a*x))/16 - a**5*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3*sqrt(c) /(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*a*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a* *2*x**2))) - sqrt(c)/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*sqrt(c)*asin(1/(a*x))/16 + I*a**5*sqrt(c)/(16*x*sqrt(1 - 1 /(a**2*x**2))) - I*a**3*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a* sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(6*a*x**7*sqrt(1 - 1 /(a**2*x**2))), True))
\[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=\int { -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{7}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (128) = 256\).
Time = 0.31 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.84 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=-\frac {3 \, a^{6} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{8 \, \sqrt {-c}} + \frac {45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{11} a^{6} c^{2} + 65 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{9} a^{6} c^{3} + 960 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{8} a^{5} \sqrt {-c} c^{3} {\left | a \right |} - 750 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{6} c^{4} - 640 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{5} \sqrt {-c} c^{4} {\left | a \right |} - 750 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{6} c^{5} + 65 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{6} c^{6} - 384 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{5} \sqrt {-c} c^{6} {\left | a \right |} + 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{6} c^{7} + 64 \, a^{5} \sqrt {-c} c^{7} {\left | a \right |}}{120 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{6}} \]
-3/8*a^6*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqr t(-c) + 1/120*(45*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^11*a^6*c^2 + 65* (sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^9*a^6*c^3 + 960*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^8*a^5*sqrt(-c)*c^3*abs(a) - 750*(sqrt(-a^2*c)*x - sq rt(-a^2*c*x^2 + c))^7*a^6*c^4 - 640*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c) )^6*a^5*sqrt(-c)*c^4*abs(a) - 750*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^ 5*a^6*c^5 + 65*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^6*c^6 - 384*(sq rt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^5*sqrt(-c)*c^6*abs(a) + 45*(sqrt( -a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^6*c^7 + 64*a^5*sqrt(-c)*c^7*abs(a))/(( sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^6
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^7} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^7\,\left (a^2\,x^2-1\right )} \,d x \]