Integrand size = 27, antiderivative size = 111 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}} \]
3/2*a*(c-c/a^2/x^2)^(1/2)+1/2*(a*x+1)*(c-c/a^2/x^2)^(1/2)/x+3/2*a^2*x*arct anh((-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 = 78, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left ((1+4 a x) \sqrt {-1+a^2 x^2}-3 a^2 x^2 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{2 x \sqrt {-1+a^2 x^2}} \]
(Sqrt[c - c/(a^2*x^2)]*((1 + 4*a*x)*Sqrt[-1 + a^2*x^2] - 3*a^2*x^2*ArcTan[ 1/Sqrt[-1 + a^2*x^2]]))/(2*x*Sqrt[-1 + a^2*x^2])
Time = 0.60 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6717, 6709, 540, 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.
| \(\displaystyle \int \frac {\sqrt {c-\frac {c}{a^2 x^2}} e^{2 \coth ^{-1}(a x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(a x+1)^2}{x^3 \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{2} \int -\frac {a (3 a x+4)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{2} \int \frac {a (3 a x+4)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{2} a \int \frac {3 a x+4}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{2} a \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{2} a \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{2} a \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {x \left (\frac {1}{2} a \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
-((Sqrt[c - c/(a^2*x^2)]*x*(-1/2*Sqrt[1 - a^2*x^2]/x^2 + (a*((-4*Sqrt[1 - a^2*x^2])/x - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/Sqrt[1 - a^2*x^2])
3.9.91.3.1 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_))^(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 = 0.61 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {\left (4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \left (a^{2} x^{2}-1\right )}-\frac {3 a^{2} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{2 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(142\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{3}+4 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x +4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c \,x^{2}+3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c \,x^{2}+a^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{2}\right )}{2 x \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c}\) | \(347\) |
1/2*(4*a^3*x^3+a^2*x^2-4*a*x-1)/x*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1 )-3/2*a^2/(-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 = 177, normalized size of antiderivative = 1.59 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\left [\frac {3 \, a \sqrt {-c} x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, -\frac {3 \, a \sqrt {c} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right ] \]
[1/4*(3*a*sqrt(-c)*x*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*(4*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x , -1/2*(3*a*sqrt(c)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/( a^2*c*x^2 - c)) - (4*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{2} \left (a x - 1\right )}\, dx \]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x - 1\right )} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (91) = 182\).
Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.75 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx={\left (3 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a c \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{2} \mathrm {sgn}\left (x\right ) - 4 \, c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a}\right )} {\left | a \right |} \]
(3*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x) - ((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*a*c*sgn(x) - 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*c^(3/2)*abs(a)*sgn(x) - (sqrt(a^2*c)*x - sqrt(a^2*c *x^2 - c))*a*c^2*sgn(x) - 4*c^(5/2)*abs(a)*sgn(x))/(((sqrt(a^2*c)*x - sqrt (a^2*c*x^2 - c))^2 + c)^2*a))*abs(a)
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{x^2\,\left (a\,x-1\right )} \,d x \]