Integrand size = 21, antiderivative size = 72 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
-a*arctanh(c^(1/2)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2))*c^(1/2)-c*(-a^2*x^ 2+1)^(1/2)/x/(-a*c*x+c)^(1/2)
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {\sqrt {c-a c x} \left (1+a x+a x \sqrt {1+a x} \text {arctanh}\left (\sqrt {1+a x}\right )\right )}{x \sqrt {1-a^2 x^2}} \]
-((Sqrt[c - a*c*x]*(1 + a*x + a*x*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 + a*x]]))/( x*Sqrt[1 - a^2*x^2]))
Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6678, 575, 573, 219}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{x^2 \sqrt {c-a c x}}dx\) |
\(\Big \downarrow \) 575 |
\(\displaystyle c \left (\frac {a \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}}dx}{2 c}-\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )\) |
\(\Big \downarrow \) 573 |
\(\displaystyle c \left (-a \int \frac {1}{1-\frac {c \left (1-a^2 x^2\right )}{c-a c x}}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle c \left (-\frac {a \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )}{\sqrt {c}}-\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )\) |
c*(-(Sqrt[1 - a^2*x^2]/(x*Sqrt[c - a*c*x])) - (a*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]])/Sqrt[c])
3.4.92.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*x)^(m + 1)*(c + d*x)^n*((a + b*x^2)^p/(e*(m + 1))), x] + Simp[b*(n/(d*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n + 1)*(a + b*x^ 2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m + p] && LeQ[m + p + 2, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x \sqrt {c}}\) | \(78\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {a \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(137\) |
(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(arctanh((c*(a*x+1))^(1/2)/c^(1/2))* a*c*x+(c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2)/x/c^(1/2)
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.88 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x^{2} - x}\right ] \]
[1/2*((a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqr t(-a*c*x + c))/(a*x^2 - x), -((a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^ 2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - sqrt(-a^2*x^2 + 1)*sqr t(-a*c*x + c))/(a*x^2 - x)]
\[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {{\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) - \sqrt {2} a^{2} \sqrt {-c}}{\sqrt {-c} \sqrt {c}} - \frac {\sqrt {a c x + c} a}{c x}\right )} c^{2}}{a {\left | c \right |}} \]
(a^2*arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) - (a^2*sqrt(c)*arctan(sqrt( 2)*sqrt(c)/sqrt(-c)) - sqrt(2)*a^2*sqrt(-c))/(sqrt(-c)*sqrt(c)) - sqrt(a*c *x + c)*a/(c*x))*c^2/(a*abs(c))
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,\left (a\,x+1\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]