Integrand size = 22, antiderivative size = 175 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)}+\frac {5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{4 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \] Output:
(-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)+5/4*(-a^2*x^2+1)^(3/2)*ln(-a*x+1)/a^4/(c- c/a^2/x^2)^(3/2)/x^3-1/4*(-a^2*x^2+1)^(3/2)*ln(a*x+1)/a^4/(c-c/a^2/x^2)^(3 /2)/x^3
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \left (-1+a^2 x^2\right ) \left (\frac {x}{a^3}+\frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (1+a x)}{4 a^4}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \] Input:
Integrate[E^ArcTanh[a*x]/(c - c/(a^2*x^2))^(3/2),x]
Output:
-((Sqrt[1 - a^2*x^2]*(-1 + a^2*x^2)*(x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[ 1 - a*x])/(4*a^4) - Log[1 + a*x]/(4*a^4)))/((c - c/(a^2*x^2))^(3/2)*x^3))
Time = 0.73 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.46, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3}{(1-a x)^2 (a x+1)}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \left (-\frac {1}{4 a^3 (a x+1)}+\frac {1}{a^3}+\frac {5}{4 a^3 (a x-1)}+\frac {1}{2 a^3 (a x-1)^2}\right )dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (a x+1)}{4 a^4}+\frac {x}{a^3}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
Input:
Int[E^ArcTanh[a*x]/(c - c/(a^2*x^2))^(3/2),x]
Output:
((1 - a^2*x^2)^(3/2)*(x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[1 - a*x])/(4*a^ 4) - Log[1 + a*x]/(4*a^4)))/((c - c/(a^2*x^2))^(3/2)*x^3)
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 = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.54
method | result | size |
default | \(\frac {\left (-4 a^{2} x^{2}+\ln \left (a x +1\right ) x a -5 a \ln \left (a x -1\right ) x +4 a x -\ln \left (a x +1\right )+5 \ln \left (a x -1\right )+2\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{4 a^{4} x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\) | \(94\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x,method=_RETURNVERBOSE )
Output:
1/4*(-4*a^2*x^2+ln(a*x+1)*x*a-5*a*ln(a*x-1)*x+4*a*x-ln(a*x+1)+5*ln(a*x-1)+ 2)*(a*x+1)*(-a^2*x^2+1)^(1/2)/a^4/x^3/(c*(a^2*x^2-1)/a^2/x^2)^(3/2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="fri cas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*a^4*x^4*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^5* c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {a x + 1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(3/2),x)
Output:
Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/( a*x)))**(3/2)), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="max ima")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(c - c/(a^2*x^2))^(3/2)), x)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="gia c")
Output:
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. \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {a\,x+1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((a*x + 1)/((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(1/2)),x)
Output:
int((a*x + 1)/((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(1/2)), x)
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.37 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, i \left (5 \,\mathrm {log}\left (a x -1\right ) a x -5 \,\mathrm {log}\left (a x -1\right )-\mathrm {log}\left (a x +1\right ) a x +\mathrm {log}\left (a x +1\right )+4 a^{2} x^{2}-6 a x \right )}{4 a \,c^{2} \left (a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x)
Output:
(sqrt(c)*i*(5*log(a*x - 1)*a*x - 5*log(a*x - 1) - log(a*x + 1)*a*x + log(a *x + 1) + 4*a**2*x**2 - 6*a*x))/(4*a*c**2*(a*x - 1))