Integrand size = 27, antiderivative size = 137 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{3 x}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}+\frac {a^3 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}} \]
a^2*(c-c/a^2/x^2)^(1/2)+1/3*a*(a*x+1)*(c-c/a^2/x^2)^(1/2)/x+1/3*(a*x+1)^2* (c-c/a^2/x^2)^(1/2)/x^2+a^3*x*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))*(c-c/a ^2/x^2)^(1/2)/(-a*x+1)^(1/2)/(a*x+1)^(1/2)
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.63 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (1+3 a x+5 a^2 x^2\right )-3 a^3 x^3 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{3 x^2 \sqrt {-1+a^2 x^2}} \]
(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1 + 3*a*x + 5*a^2*x^2) - 3*a^3 *x^3*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(3*x^2*Sqrt[-1 + a^2*x^2])
Time = 0.61 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6717, 6709, 540, 25, 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.
| \(\displaystyle \int \frac {\sqrt {c-\frac {c}{a^2 x^2}} e^{2 \coth ^{-1}(a x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(a x+1)^2}{x^4 \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{3} \int -\frac {a (5 a x+6)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} \int \frac {a (5 a x+6)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \int \frac {5 a x+6}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \left (-\frac {1}{2} \int -\frac {2 a (3 a x+5)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \left (a \int \frac {3 a x+5}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \left (a \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \left (a \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {5 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} a \left (a \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {5 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {x \left (\frac {1}{3} a \left (a \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {5 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
-((Sqrt[c - c/(a^2*x^2)]*x*(-1/3*Sqrt[1 - a^2*x^2]/x^3 + (a*((-3*Sqrt[1 - a^2*x^2])/x^2 + a*((-5*Sqrt[1 - a^2*x^2])/x - 3*a*ArcTanh[Sqrt[1 - a^2*x^2 ]])))/3))/Sqrt[1 - a^2*x^2])
3.9.92.3.1 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[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]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {\left (5 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-3 a x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{3 x^{2} \left (a^{2} x^{2}-1\right )}-\frac {a^{3} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{\sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(151\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a \left (-6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{4}+6 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x^{2}+6 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} a \,x^{3}-6 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c \,x^{3}-6 \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} a \,x^{3}+3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c \,x^{3}+3 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{2} x +3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{3}+a {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{3 x^{2} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c}\) | \(378\) |
1/3*(5*a^4*x^4+3*a^3*x^3-4*a^2*x^2-3*a*x-1)/x^2*(c*(a^2*x^2-1)/a^2/x^2)^(1 /2)/(a^2*x^2-1)-a^3/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*(a^2*c*x^2-c)^(1/2))/ x)*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)/(a^2*x^2-1)*x
Time = 0.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.47 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\left [\frac {3 \, a^{2} \sqrt {-c} x^{2} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, -\frac {3 \, a^{2} \sqrt {c} x^{2} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\right ] \]
[1/6*(3*a^2*sqrt(-c)*x^2*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*(5*a^2*x^2 + 3*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^2, -1/3*(3*a^2*sqrt(c)*x^2*arctan(a*sqrt(c)*x*sqrt((a^2*c *x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (5*a^2*x^2 + 3*a*x + 1)*sqrt((a^2* c*x^2 - c)/(a^2*x^2)))/x^2]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{3} \left (a x - 1\right )}\, dx \]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x - 1\right )} x^{3}} \,d x } \]
Time = 0.69 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.69 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\frac {2}{3} \, {\left (3 \, a \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a c \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 12 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{3} \mathrm {sgn}\left (x\right ) - 5 \, c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{3}}\right )} {\left | a \right |} \]
2/3*(3*a*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sg n(x) - (3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*a*c*sgn(x) - 3*(sqrt(a^2 *c)*x - sqrt(a^2*c*x^2 - c))^4*c^(3/2)*abs(a)*sgn(x) - 12*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*c^(5/2)*abs(a)*sgn(x) - 3*(sqrt(a^2*c)*x - sqrt(a^ 2*c*x^2 - c))*a*c^3*sgn(x) - 5*c^(7/2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sq rt(a^2*c*x^2 - c))^2 + c)^3)*abs(a)
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{x^3\,\left (a\,x-1\right )} \,d x \]