Integrand size = 27, antiderivative size = 149 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{5 c^2}-\frac {a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{c^2 \left (a+\frac {1}{x}\right )^2}-\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 c x}-\frac {3}{4} a^4 \sqrt {c} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right ) \] Output:
-1/5*a^4*(c-c/a^2/x^2)^(5/2)/c^2-a^6*(c-c/a^2/x^2)^(5/2)/c^2/(a+1/x)^2-3/4 *a^3*(c-c/a^2/x^2)^(1/2)/x-1/2*a^3*(c-c/a^2/x^2)^(3/2)/c/x-3/4*a^4*c^(1/2) *arctan(c^(1/2)/a/(c-c/a^2/x^2)^(1/2)/x)
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (4-10 a x+12 a^2 x^2-15 a^3 x^3+24 a^4 x^4\right )+15 a^5 x^5 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{20 x^4 \sqrt {-1+a^2 x^2}} \] Input:
Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^5),x]
Output:
-1/20*(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(4 - 10*a*x + 12*a^2*x^2 - 15*a^3*x^3 + 24*a^4*x^4) + 15*a^5*x^5*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(x^ 4*Sqrt[-1 + a^2*x^2])
Time = 0.86 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6709, 570, 540, 27, 539, 27, 539, 27, 539, 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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^6 (a x+1)^2}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x^6 \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} \int \frac {a (10-9 a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \int \frac {10-9 a x}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {1}{4} \int \frac {6 a (6-5 a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \int \frac {6-5 a x}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-\frac {1}{3} \int \frac {3 a (5-4 a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \int \frac {5-4 a x}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} \int \frac {a (8-5 a x)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} a \int \frac {8-5 a x}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} a \left (-5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} a \left (-\frac {5}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} a \left (\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \left (-\frac {1}{5} a \left (-\frac {3}{2} a \left (-a \left (-\frac {1}{2} a \left (5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x^3}\right )-\frac {5 \sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^5),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/5*Sqrt[1 - a^2*x^2]/x^5 - (a*((-5*Sqrt[1 - a^ 2*x^2])/(2*x^4) - (3*a*((-2*Sqrt[1 - a^2*x^2])/x^3 - a*((-5*Sqrt[1 - a^2*x ^2])/(2*x^2) - (a*((-8*Sqrt[1 - a^2*x^2])/x + 5*a*ArcTanh[Sqrt[1 - a^2*x^2 ]]))/2)))/2))/5))/Sqrt[1 - a^2*x^2]
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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {\left (24 x^{6} a^{6}-15 a^{5} x^{5}-12 a^{4} x^{4}+5 a^{3} x^{3}-8 a^{2} x^{2}+10 a x -4\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{20 x^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 a^{5} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{4 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(167\) |
| default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-40 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{4} c \,x^{6}+40 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{4} x^{4}-15 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c \,x^{5}+40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{2} x^{5}-40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) a^{2} x^{5}+40 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c \,x^{5}-25 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{3} x^{3}-15 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) a \,c^{2} x^{5}+16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{2} x^{2}-10 a {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} x \sqrt {-\frac {c}{a^{2}}}+4 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{20 x^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \sqrt {-\frac {c}{a^{2}}}}\) | \(447\) |
Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x,method=_RETURNVERBOSE )
Output:
-1/20*(24*a^6*x^6-15*a^5*x^5-12*a^4*x^4+5*a^3*x^3-8*a^2*x^2+10*a*x-4)/x^4* (c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)-3/4*a^5/(-c)^(1/2)*ln((-2*c+2*(- c)^(1/2)*(a^2*c*x^2-c)^(1/2))/x)*x*(c*(a^2*x^2-1))^(1/2)*(c*(a^2*x^2-1)/a^ 2/x^2)^(1/2)/(a^2*x^2-1)
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\left [\frac {15 \, a^{4} \sqrt {-c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, \frac {15 \, a^{4} \sqrt {c} x^{4} \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) - {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right ] \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="fri cas")
Output:
[1/40*(15*a^4*sqrt(-c)*x^4*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^ 2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(24*a^4*x^4 - 15*a^3*x^3 + 12*a^2*x^2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4, 1/20*(15*a^4*sqrt(c)*x^4 *arctan(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/sqrt(c)) - (24*a^4*x^4 - 15*a^ 3*x^3 + 12*a^2*x^2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=- \int \left (- \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\, dx \] Input:
integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)
Output:
-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**6 + x**5), x) - Integral(a*x*sqrt (c - c/(a**2*x**2))/(a*x**6 + x**5), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{2} x^{5}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="max ima")
Output:
-integrate((a^2*x^2 - 1)*sqrt(c - c/(a^2*x^2))/((a*x + 1)^2*x^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (127) = 254\).
Time = 0.26 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.43 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {1}{10} \, {\left (15 \, a^{3} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm {sgn}\left (x\right ) + 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 200 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm {sgn}\left (x\right ) + 120 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm {sgn}\left (x\right ) + 24 \, a^{2} c^{\frac {11}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\right )} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="gia c")
Output:
1/10*(15*a^3*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c) )*sgn(x) - (15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^9*a^3*c*sgn(x) + 70*( sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*a^3*c^2*sgn(x) + 40*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a^2*c^(5/2)*abs(a)*sgn(x) + 200*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a^2*c^(7/2)*abs(a)*sgn(x) - 70*(sqrt(a^2*c)*x - sqr t(a^2*c*x^2 - c))^3*a^3*c^4*sgn(x) + 120*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a^2*c^(9/2)*abs(a)*sgn(x) - 15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c) )*a^3*c^5*sgn(x) + 24*a^2*c^(11/2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a ^2*c*x^2 - c))^2 + c)^5)*abs(a)
Timed out. \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{x^5\,{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-((c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(x^5*(a*x + 1)^2),x)
Output:
-int(((c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(x^5*(a*x + 1)^2), x)
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c}\, \left (30 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{5} x^{5}-24 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+15 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-12 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+10 \sqrt {a^{2} x^{2}-1}\, a x -4 \sqrt {a^{2} x^{2}-1}+24 a^{5} x^{5}\right )}{20 a \,x^{5}} \] Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x)
Output:
(sqrt(c)*(30*atan(sqrt(a**2*x**2 - 1) + a*x)*a**5*x**5 - 24*sqrt(a**2*x**2 - 1)*a**4*x**4 + 15*sqrt(a**2*x**2 - 1)*a**3*x**3 - 12*sqrt(a**2*x**2 - 1 )*a**2*x**2 + 10*sqrt(a**2*x**2 - 1)*a*x - 4*sqrt(a**2*x**2 - 1) + 24*a**5 *x**5))/(20*a*x**5)