Integrand size = 24, antiderivative size = 170 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)^2}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1+a x)}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]
(-a^2*x^2+1)^(3/2)/a^3/(c-c/a^2/x^2)^(3/2)/x^2+1/2*(-a^2*x^2+1)^(3/2)/a^4/ (c-c/a^2/x^2)^(3/2)/x^3/(a*x+1)^2-3*(-a^2*x^2+1)^(3/2)/a^4/(c-c/a^2/x^2)^( 3/2)/x^3/(a*x+1)-3*(-a^2*x^2+1)^(3/2)*ln(a*x+1)/a^4/(c-c/a^2/x^2)^(3/2)/x^ 3
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \left (-1+a^2 x^2\right ) \left (\frac {x}{a^3}+\frac {1}{2 a^4 (1+a x)^2}-\frac {3}{a^4 (1+a x)}-\frac {3 \log (1+a x)}{a^4}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]
-((Sqrt[1 - a^2*x^2]*(-1 + a^2*x^2)*(x/a^3 + 1/(2*a^4*(1 + a*x)^2) - 3/(a^ 4*(1 + a*x)) - (3*Log[1 + a*x])/a^4))/((c - c/(a^2*x^2))^(3/2)*x^3))
Time = 0.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6710, 6700, 49, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {e^{-3 \text {arctanh}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3}{(a x+1)^3}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \left (\frac {1}{a^3}-\frac {3}{a^3 (a x+1)}+\frac {3}{a^3 (a x+1)^2}-\frac {1}{a^3 (a x+1)^3}\right )dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (-\frac {3}{a^4 (a x+1)}+\frac {1}{2 a^4 (a x+1)^2}-\frac {3 \log (a x+1)}{a^4}+\frac {x}{a^3}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
((1 - a^2*x^2)^(3/2)*(x/a^3 + 1/(2*a^4*(1 + a*x)^2) - 3/(a^4*(1 + a*x)) - (3*Log[1 + a*x])/a^4))/((c - c/(a^2*x^2))^(3/2)*x^3)
3.8.38.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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 = 106, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {\left (-2 a^{3} x^{3}+6 a^{2} \ln \left (a x +1\right ) x^{2}-4 a^{2} x^{2}+12 a \ln \left (a x +1\right ) x +4 a x +6 \ln \left (a x +1\right )+5\right ) \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a x +1\right ) a^{4} x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\) | \(106\) |
1/2*(-2*a^3*x^3+6*a^2*ln(a*x+1)*x^2-4*a^2*x^2+12*a*ln(a*x+1)*x+4*a*x+6*ln( a*x+1)+5)*(a*x-1)*(-a^2*x^2+1)^(1/2)/(a*x+1)/a^4/x^3/(c*(a^2*x^2-1)/a^2/x^ 2)^(3/2)
Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.81 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) + {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac {6 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) - {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
[-1/2*(3*(a^4*x^4 + 2*a^3*x^3 - 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^4*x^4 + 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^2*x^ 4 + 2*a^4*c^2*x^3 - 2*a^2*c^2*x - a*c^2), 1/2*(6*(a^4*x^4 + 2*a^3*x^3 - 2* a*x - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*s qrt((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)) - (2*a^4*x^4 + 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c )/(a^2*x^2)))/(a^5*c^2*x^4 + 2*a^4*c^2*x^3 - 2*a^2*c^2*x - a*c^2)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}\, dx \]
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))* *(3/2)*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^3} \,d x \]