Integrand size = 27, antiderivative size = 78 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-\sqrt {c-a^2 c x^2}-2 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
-2*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))*c^(1/2)-arctanh((-a^2*c*x^2+c) ^(1/2)/c^(1/2))*c^(1/2)-(-a^2*c*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-\sqrt {c-a^2 c x^2}+2 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )+\sqrt {c} \log (x)-\sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \]
-Sqrt[c - a^2*c*x^2] + 2*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c] *(-1 + a^2*x^2))] + Sqrt[c]*Log[x] - Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2* c*x^2]]
Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6702, 541, 25, 27, 538, 224, 216, 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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx\) |
| \(\Big \downarrow \) 6702 |
| \(\displaystyle c \int \frac {(1-a x)^2}{x \sqrt {c-a^2 c x^2}}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -\frac {a^2 c (1-2 a x)}{x \sqrt {c-a^2 c x^2}}dx}{a^2 c}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\int \frac {a^2 c (1-2 a x)}{x \sqrt {c-a^2 c x^2}}dx}{a^2 c}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\int \frac {1-2 a x}{x \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle c \left (-2 a \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-2 a \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (-\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
c*(-(Sqrt[c - a^2*c*x^2]/c) - (2*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] )/Sqrt[c] - ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]/Sqrt[c])
3.13.40.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/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]) && ILtQ[n/2, 0]
Time = 0.47 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )-2 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\) | \(120\) |
(-a^2*c*x^2+c)^(1/2)-c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-2* (-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2)-2*a*c/(a^2*c)^(1/2)*arctan((a^2*c)^ (1/2)*x/(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2))
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.51 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\left [2 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c}, -\sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - \sqrt {-a^{2} c x^{2} + c}\right ] \]
[2*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 1/2* sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqr t(-a^2*c*x^2 + c), -sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x ^2 - c)) + sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - sqrt(-a^2*c*x^2 + c)]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=- \int \left (- \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{2} + x}\right )\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{2} + x}\, dx \]
-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**2 + x), x) - Integral(a*x*sqrt(-a* *2*c*x**2 + c)/(a*x**2 + x), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int { -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.59 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-{\left (\frac {{\left (a x + 1\right )} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{a^{2}} - \frac {2 \, c \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{a^{2} \sqrt {-c}} - \frac {4 \, \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{a^{2}}\right )} a {\left | a \right |} \]
-((a*x + 1)*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a)/a^2 - 2*c*arc tan(sqrt(-c + 2*c/(a*x + 1))/sqrt(-c))*sgn(1/(a*x + 1))*sgn(a)/(a^2*sqrt(- c)) - 4*sqrt(c)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a*x + 1))* sgn(a)/a^2)*a*abs(a)
Timed out. \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{x\,{\left (a\,x+1\right )}^2} \,d x \]