Integrand size = 24, antiderivative size = 269 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\left (1-a^2 x^2\right )^{5/2}}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}+\frac {\left (1-a^2 x^2\right )^{5/2}}{6 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)^3}-\frac {9 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)^2}+\frac {31 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)}+\frac {49 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {\left (1-a^2 x^2\right )^{5/2} \log (1+a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \]
(-a^2*x^2+1)^(5/2)/a^5/(c-c/a^2/x^2)^(5/2)/x^4+1/6*(-a^2*x^2+1)^(5/2)/a^6/ (c-c/a^2/x^2)^(5/2)/x^5/(-a*x+1)^3-9/8*(-a^2*x^2+1)^(5/2)/a^6/(c-c/a^2/x^2 )^(5/2)/x^5/(-a*x+1)^2+31/8*(-a^2*x^2+1)^(5/2)/a^6/(c-c/a^2/x^2)^(5/2)/x^5 /(-a*x+1)+49/16*(-a^2*x^2+1)^(5/2)*ln(-a*x+1)/a^6/(c-c/a^2/x^2)^(5/2)/x^5- 1/16*(-a^2*x^2+1)^(5/2)*ln(a*x+1)/a^6/(c-c/a^2/x^2)^(5/2)/x^5
Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 \left (-70+135 a x-21 a^2 x^2-72 a^3 x^3+24 a^4 x^4\right )+147 (-1+a x)^3 \log (1-a x)-3 (-1+a x)^3 \log (1+a x)\right )}{48 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^3} \]
(Sqrt[1 - a^2*x^2]*(2*(-70 + 135*a*x - 21*a^2*x^2 - 72*a^3*x^3 + 24*a^4*x^ 4) + 147*(-1 + a*x)^3*Log[1 - a*x] - 3*(-1 + a*x)^3*Log[1 + a*x]))/(48*a^2 *c^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)^3)
Time = 0.49 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6710, 6700, 99, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {e^{3 \text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {x^5}{(1-a x)^4 (a x+1)}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \left (-\frac {1}{16 a^5 (a x+1)}+\frac {1}{a^5}+\frac {49}{16 a^5 (a x-1)}+\frac {31}{8 a^5 (a x-1)^2}+\frac {9}{4 a^5 (a x-1)^3}+\frac {1}{2 a^5 (a x-1)^4}\right )dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {31}{8 a^6 (1-a x)}-\frac {9}{8 a^6 (1-a x)^2}+\frac {1}{6 a^6 (1-a x)^3}+\frac {49 \log (1-a x)}{16 a^6}-\frac {\log (a x+1)}{16 a^6}+\frac {x}{a^5}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
((1 - a^2*x^2)^(5/2)*(x/a^5 + 1/(6*a^6*(1 - a*x)^3) - 9/(8*a^6*(1 - a*x)^2 ) + 31/(8*a^6*(1 - a*x)) + (49*Log[1 - a*x])/(16*a^6) - Log[1 + a*x]/(16*a ^6)))/((c - c/(a^2*x^2))^(5/2)*x^5)
3.8.12.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {\left (-48 a^{4} x^{4}+3 a^{3} \ln \left (a x +1\right ) x^{3}-147 a^{3} \ln \left (a x -1\right ) x^{3}+144 a^{3} x^{3}-9 a^{2} \ln \left (a x +1\right ) x^{2}+441 a^{2} \ln \left (a x -1\right ) x^{2}+42 a^{2} x^{2}+9 a \ln \left (a x +1\right ) x -441 a \ln \left (a x -1\right ) x -270 a x -3 \ln \left (a x +1\right )+147 \ln \left (a x -1\right )+140\right ) \left (a x +1\right )^{2} \sqrt {-a^{2} x^{2}+1}}{48 \left (a x -1\right ) a^{6} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\) | \(176\) |
-1/48*(-48*a^4*x^4+3*a^3*ln(a*x+1)*x^3-147*a^3*ln(a*x-1)*x^3+144*a^3*x^3-9 *a^2*ln(a*x+1)*x^2+441*a^2*ln(a*x-1)*x^2+42*a^2*x^2+9*a*ln(a*x+1)*x-441*a* ln(a*x-1)*x-270*a*x-3*ln(a*x+1)+147*ln(a*x-1)+140)*(a*x+1)^2*(-a^2*x^2+1)^ (1/2)/(a*x-1)/a^6/x^5/(c*(a^2*x^2-1)/a^2/x^2)^(5/2)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c ^3*x^7 - 3*a^6*c^3*x^6 + a^5*c^3*x^5 + 5*a^4*c^3*x^4 - 5*a^3*c^3*x^3 - a^2 *c^3*x^2 + 3*a*c^3*x - c^3), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}}}\, dx \]
Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]