Integrand size = 24, antiderivative size = 174 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)^2}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \] 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)^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 = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.50 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (5-4 a x-4 a^2 x^2+2 a^3 x^3+6 (-1+a x)^2 \log (1-a x)\right )}{2 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^2} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(5 - 4*a*x - 4*a^2*x^2 + 2*a^3*x^3 + 6*(-1 + a*x)^2*Log [1 - a*x]))/(2*a^2*c*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)^2)
Time = 0.75 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.46, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3}{(1-a x)^3}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (-\frac {3}{a^4 (1-a x)}+\frac {1}{2 a^4 (1-a x)^2}-\frac {3 \log (1-a x)}{a^4}-\frac {x}{a^3}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
Input:
Int[E^(3*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)^2) - 3/(a^4*(1 - a*x)) - (3*Log[1 - a*x])/a^4))/((c - c/(a^2*x^2))^(3/2)*x^3)
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 = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\left (2 a^{3} x^{3}+6 a^{2} \ln \left (a x -1\right ) x^{2}-4 a^{2} x^{2}-12 a \ln \left (a x -1\right ) x -4 a x +6 \ln \left (a x -1\right )+5\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a x -1\right ) a^{4} x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\) | \(106\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x,method=_RETURNVERBO SE)
Output:
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 = 0.21 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.74 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) - {\left (2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} - 2 \, a^{4} c^{2} x^{3} + 2 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac {6 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right ) + {\left (2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{5} c^{2} x^{4} - 2 \, a^{4} c^{2} x^{3} + 2 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="f ricas")
Output:
[-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)]
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(3/2),x)
Output:
Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="m axima")
Output:
integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(3/2)), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x, algorithm="g iac")
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. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(3/2)),x)
Output:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(3/2)), x)
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.44 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, i \left (-6 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+12 \,\mathrm {log}\left (a x -1\right ) a x -6 \,\mathrm {log}\left (a x -1\right )-2 a^{3} x^{3}+6 a^{2} x^{2}-3\right )}{2 a \,c^{2} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x)
Output:
(sqrt(c)*i*( - 6*log(a*x - 1)*a**2*x**2 + 12*log(a*x - 1)*a*x - 6*log(a*x - 1) - 2*a**3*x**3 + 6*a**2*x**2 - 3))/(2*a*c**2*(a**2*x**2 - 2*a*x + 1))