Integrand size = 24, antiderivative size = 122 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)}+\frac {3 \sqrt {1-a^2 x^2} \log (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]
-(-a^2*x^2+1)^(1/2)/a/(c-c/a^2/x^2)^(1/2)+2*(-a^2*x^2+1)^(1/2)/a^2/x/(a*x+ 1)/(c-c/a^2/x^2)^(1/2)+3*ln(a*x+1)*(-a^2*x^2+1)^(1/2)/a^2/x/(c-c/a^2/x^2)^ (1/2)
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-a x+\frac {2}{1+a x}+3 \log (1+a x)\right )}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]
Time = 0.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6710, 6700, 86, 2009}
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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \text {arctanh}(a x)} x}{\sqrt {1-a^2 x^2}}dx}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x (1-a x)}{(a x+1)^2}dx}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (\frac {3}{(a x+1) a}-\frac {2}{(a x+1)^2 a}-\frac {1}{a}\right )dx}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {2}{a^2 (a x+1)}+\frac {3 \log (a x+1)}{a^2}-\frac {x}{a}\right )}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
(Sqrt[1 - a^2*x^2]*(-(x/a) + 2/(a^2*(1 + a*x)) + (3*Log[1 + a*x])/a^2))/(S qrt[c - c/(a^2*x^2)]*x)
3.8.37.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
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/x^(2*p))*(1 - a ^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && !IntegerQ[n/2]
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} x^{2}+3 a \ln \left (a x +1\right ) x -a x +3 \ln \left (a x +1\right )+2\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2} \left (a x +1\right )}\) | \(78\) |
1/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x*(-a^2*x^2+1)^(1/2)/a^2*(-a^2*x^2+3*a*ln( a*x+1)*x-a*x+3*ln(a*x+1)+2)/(a*x+1)
Time = 0.29 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.60 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\left [-\frac {3 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) + 2 \, {\left (a^{3} x^{3} + 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c\right )}}, \frac {3 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) - {\left (a^{3} x^{3} + 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c}\right ] \]
[-1/2*(3*(a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x ^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^ 3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) + 2*(a^3*x^3 + 3*a^2*x^2)*sqrt( -a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 + a^3*c*x^2 - a^ 2*c*x - a*c), (3*(a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(c)*arctan((a^2*x^2 + 2 *a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3* c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) - (a^3*x^3 + 3*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 + a^3*c*x^2 - a^2*c*x - a *c)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}\, dx \]
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x )))*(a*x + 1)**3), x)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=-\frac {i \, a^{2} \sqrt {c} x^{2} + i \, a \sqrt {c} x - 2 i \, \sqrt {c}}{a^{2} c x + a c} + \frac {3 i \, \log \left (a x + 1\right )}{a \sqrt {c}} \]
-(I*a^2*sqrt(c)*x^2 + I*a*sqrt(c)*x - 2*I*sqrt(c))/(a^2*c*x + a*c) + 3*I*l og(a*x + 1)/(a*sqrt(c))
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}} \,d x } \]
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3} \,d x \]