Integrand size = 24, antiderivative size = 123 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}+\frac {3 \sqrt {1-a^2 x^2} \log (1-a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \] Output:
(-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/(c-c/a^2 /x^2)^(1/2)/x/(-a*x+1)+3*(-a^2*x^2+1)^(1/2)*ln(-a*x+1)/a^2/(c-c/a^2/x^2)^( 1/2)/x
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (a x+\frac {2}{1-a x}+3 \log (1-a x)\right )}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \] Input:
Integrate[E^(3*ArcTanh[a*x])/Sqrt[c - c/(a^2*x^2)],x]
Output:
(Sqrt[1 - a^2*x^2]*(a*x + 2/(1 - a*x) + 3*Log[1 - a*x]))/(a^2*Sqrt[c - c/( a^2*x^2)]*x)
Time = 0.45 (sec) , antiderivative size = 64, 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 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x (a x+1)}{(1-a x)^2}dx}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {2}{a^2 (1-a x)}+\frac {3 \log (1-a x)}{a^2}+\frac {x}{a}\right )}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
Input:
Int[E^(3*ArcTanh[a*x])/Sqrt[c - c/(a^2*x^2)],x]
Output:
(Sqrt[1 - a^2*x^2]*(x/a + 2/(a^2*(1 - a*x)) + (3*Log[1 - a*x])/a^2))/(Sqrt [c - c/(a^2*x^2)]*x)
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 = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}+3 a \ln \left (a x -1\right ) x -a x -3 \ln \left (a x -1\right )-2\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2} \left (a x -1\right )}\) | \(77\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x,method=_RETURNVERBO SE)
Output:
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 = 0.12 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.58 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\left [-\frac {3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) - 2 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c\right )}}, \frac {3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right ) + {\left (a^{3} x^{3} - 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x, algorithm="f ricas")
Output:
[-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)]
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(1/2),x)
Output:
Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x) )*(1 + 1/(a*x)))), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x, algorithm="m axima")
Output:
-1/2*I/((a*x + 1)*(a*x - 1)*a*sqrt(c)) - integrate((a^4*x^4 + 3*a^3*x^3 + 3*a^2*x^2)/((I*a^2*sqrt(c)*x^2 - I*sqrt(c))*(a*x + 1)*(a*x - 1)), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),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. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {{\left (a\,x+1\right )}^3}{\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2)),x)
Output:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.39 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {c}\, i \left (-3 \,\mathrm {log}\left (a x -1\right ) a x +3 \,\mathrm {log}\left (a x -1\right )-a^{2} x^{2}+3 a x \right )}{a c \left (a x -1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x)
Output:
(sqrt(c)*i*( - 3*log(a*x - 1)*a*x + 3*log(a*x - 1) - a**2*x**2 + 3*a*x))/( a*c*(a*x - 1))