Integrand size = 27, antiderivative size = 133 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {7}{8} a^3 \sqrt {c} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right ) \] Output:
-4/3*a^3*(c-c/a^2/x^2)^(1/2)-1/4*(c-c/a^2/x^2)^(1/2)/x^3-2/3*a*(c-c/a^2/x^ 2)^(1/2)/x^2-7/8*a^2*(c-c/a^2/x^2)^(1/2)/x+7/8*a^3*c^(1/2)*arctan(c^(1/2)/ a/(c-c/a^2/x^2)^(1/2)/x)
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.71 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\sqrt {-1+a^2 x^2} \left (6+16 a x+21 a^2 x^2+32 a^3 x^3\right )+21 a^4 x^4 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{24 x^3 \sqrt {-1+a^2 x^2}} \] Input:
Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*(-(Sqrt[-1 + a^2*x^2]*(6 + 16*a*x + 21*a^2*x^2 + 32 *a^3*x^3)) + 21*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(24*x^3*Sqrt[-1 + a ^2*x^2])
Time = 0.50 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6709, 540, 25, 27, 539, 25, 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.
| \(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(a x+1)^2}{x^5 \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{4} \int -\frac {a (7 a x+8)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \int \frac {a (7 a x+8)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \int \frac {7 a x+8}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (-\frac {1}{3} \int -\frac {a (16 a x+21)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} \int \frac {a (16 a x+21)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \int \frac {16 a x+21}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (-\frac {1}{2} \int -\frac {a (21 a x+32)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} \int \frac {a (21 a x+32)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \int \frac {21 a x+32}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (21 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {32 \sqrt {1-a^2 x^2}}{x}\right )-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (\frac {21}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {32 \sqrt {1-a^2 x^2}}{x}\right )-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-\frac {21 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {32 \sqrt {1-a^2 x^2}}{x}\right )-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-21 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {32 \sqrt {1-a^2 x^2}}{x}\right )-\frac {21 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/4*Sqrt[1 - a^2*x^2]/x^4 + (a*((-8*Sqrt[1 - a^ 2*x^2])/(3*x^3) + (a*((-21*Sqrt[1 - a^2*x^2])/(2*x^2) + (a*((-32*Sqrt[1 - a^2*x^2])/x - 21*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3))/4))/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^(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 = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(-\frac {\left (32 a^{5} x^{5}+21 a^{4} x^{4}-16 a^{3} x^{3}-15 a^{2} x^{2}-16 a x -6\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 x^{3} \left (a^{2} x^{2}-1\right )}+\frac {7 a^{4} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{8 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(159\) |
| default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-48 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{5}+48 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x^{3}+48 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} a \,x^{4}-48 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c \,x^{4}-48 \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} a \,x^{4}+21 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c \,x^{4}+27 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{2} x^{2}+21 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{4}+16 a {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} x \sqrt {-\frac {c}{a^{2}}}+6 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{24 x^{3} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c}\) | \(410\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x,method=_RETURNVERBOSE )
Output:
-1/24*(32*a^5*x^5+21*a^4*x^4-16*a^3*x^3-15*a^2*x^2-16*a*x-6)/x^3*(c*(a^2*x ^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)+7/8*a^4/(-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 = 0.10 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.52 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\left [\frac {21 \, a^{3} \sqrt {-c} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, -\frac {21 \, a^{3} \sqrt {c} x^{3} \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) + {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="fri cas")
Output:
[1/48*(21*a^3*sqrt(-c)*x^3*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*(32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*s qrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3, -1/24*(21*a^3*sqrt(c)*x^3*arctan(a*x* sqrt((a^2*c*x^2 - c)/(a^2*x^2))/sqrt(c)) + (32*a^3*x^3 + 21*a^2*x^2 + 16*a *x + 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3]
\[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=- \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**4,x)
Output:
-Integral(sqrt(c - c/(a**2*x**2))/(a*x**5 - x**4), x) - Integral(a*x*sqrt( c - c/(a**2*x**2))/(a*x**5 - x**4), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a^{2} x^{2} - 1\right )} x^{4}} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="max ima")
Output:
-integrate((a*x + 1)^2*sqrt(c - c/(a^2*x^2))/((a^2*x^2 - 1)*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (109) = 218\).
Time = 0.17 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.38 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\frac {1}{12} \, {\left (21 \, a^{2} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{2} c \mathrm {sgn}\left (x\right ) + 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a^{2} c^{2} \mathrm {sgn}\left (x\right ) - 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{2} c^{3} \mathrm {sgn}\left (x\right ) - 128 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{2} c^{4} \mathrm {sgn}\left (x\right ) - 32 \, a c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4}}\right )} {\left | a \right |} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="gia c")
Output:
-1/12*(21*a^2*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c ))*sgn(x) - (21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*a^2*c*sgn(x) + 45* (sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*a^2*c^2*sgn(x) - 96*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(5/2)*abs(a)*sgn(x) - 45*(sqrt(a^2*c)*x - sq rt(a^2*c*x^2 - c))^3*a^2*c^3*sgn(x) - 128*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(7/2)*abs(a)*sgn(x) - 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c)) *a^2*c^4*sgn(x) - 32*a*c^(9/2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a^2*c *x^2 - c))^2 + c)^4)*abs(a)
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \] Input:
int(-((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)
Output:
-int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)), x)
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {\sqrt {c}\, \left (-18 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{4} x^{4}-12 \mathit {atan} \left (\frac {\sqrt {a^{2} x^{2}-1}\, a x +a^{2} x^{2}-1}{\sqrt {a^{2} x^{2}-1}+a x}\right ) a^{4} x^{4}-32 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-21 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-16 \sqrt {a^{2} x^{2}-1}\, a x -6 \sqrt {a^{2} x^{2}-1}+32 a^{4} x^{4}\right )}{24 a \,x^{4}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x)
Output:
(sqrt(c)*( - 18*atan(sqrt(a**2*x**2 - 1) + a*x)*a**4*x**4 - 12*atan((sqrt( a**2*x**2 - 1)*a*x + a**2*x**2 - 1)/(sqrt(a**2*x**2 - 1) + a*x))*a**4*x**4 - 32*sqrt(a**2*x**2 - 1)*a**3*x**3 - 21*sqrt(a**2*x**2 - 1)*a**2*x**2 - 1 6*sqrt(a**2*x**2 - 1)*a*x - 6*sqrt(a**2*x**2 - 1) + 32*a**4*x**4))/(24*a*x **4)