Integrand size = 27, antiderivative size = 75 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\sqrt {c-a^2 c x^2}-2 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
-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 = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\sqrt {c-a^2 c x^2}+2 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )-\sqrt {c} \log (x)+\sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \]
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 = 0.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6717, 6701, 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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2} e^{2 \coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x}dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle -c \int \frac {(a x+1)^2}{x \sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle -c \left (-\frac {\int -\frac {a^2 c (2 a x+1)}{x \sqrt {c-a^2 c x^2}}dx}{a^2 c}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\int \frac {a^2 c (2 a x+1)}{x \sqrt {c-a^2 c x^2}}dx}{a^2 c}-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\int \frac {2 a x+1}{x \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{c}\right )\) |
\(\Big \downarrow \) 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]))
3.7.77.3.1 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[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]) && IGtQ[n/2, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(61)=122\).
Time = 0.63 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\sqrt {-a^{2} c \,x^{2}+c}+\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )+2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\) | \(129\) |
-(-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 = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.55 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\left [2 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + \sqrt {-a^{2} c x^{2} + c}, \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + \sqrt {-a^{2} c x^{2} + c}\right ] \]
[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)]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x \left (a x - 1\right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-a^{2} {\left (\frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} - \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{2}}\right )} - a {\left (\frac {\sqrt {c} \arcsin \left (a x\right )}{a} - \frac {\sqrt {c} \log \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c}}{{\left | x \right |}} + \frac {2 \, c}{{\left | x \right |}}\right )}{a}\right )} \]
-a^2*(sqrt(c)*arcsin(a*x)/a^2 - sqrt(-a^2*c*x^2 + c)/a^2) - a*(sqrt(c)*arc sin(a*x)/a - sqrt(c)*log(2*sqrt(-a^2*c*x^2 + c)*sqrt(c)/abs(x) + 2*c/abs(x ))/a)
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.27 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=-\frac {2 \, c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, a \sqrt {-c} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \sqrt {-a^{2} c x^{2} + c} \]
-2*c*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) - 2*a*sqrt(-c)*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/abs(a) + sqr t(-a^2*c*x^2 + c)
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{x\,\left (a\,x-1\right )} \,d x \]