Integrand size = 27, antiderivative size = 147 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1+a x)}{\sqrt {1-a^2 x^2}} \] Output:
3*a*(c-c/a^2/x^2)^(1/2)/(-a^2*x^2+1)^(1/2)-1/2*(c-c/a^2/x^2)^(1/2)/x/(-a^2 *x^2+1)^(1/2)+4*a^2*(c-c/a^2/x^2)^(1/2)*x*ln(x)/(-a^2*x^2+1)^(1/2)-4*a^2*( c-c/a^2/x^2)^(1/2)*x*ln(a*x+1)/(-a^2*x^2+1)^(1/2)
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (-\frac {1}{2 x^2}+\frac {3 a}{x}+4 a^2 \log (x)-4 a^2 \log (1+a x)\right )}{\sqrt {1-a^2 x^2}} \] Input:
Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/2*1/x^2 + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[ 1 + a*x]))/Sqrt[1 - a^2*x^2]
Time = 0.82 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {1-a^2 x^2}}{x^3}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x^3 (a x+1)}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \left (-\frac {4 a^3}{a x+1}+\frac {4 a^2}{x}-\frac {3 a}{x^2}+\frac {1}{x^3}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/2*1/x^2 + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[ 1 + a*x]))/Sqrt[1 - a^2*x^2]
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 = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.46
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (8 \ln \left (a x +1\right ) x^{2} a^{2}-8 a^{2} \ln \left (x \right ) x^{2}-6 a x +1\right )}{2 x \sqrt {-a^{2} x^{2}+1}}\) | \(67\) |
Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x,method=_RETURNV ERBOSE)
Output:
-1/2*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x*(8*ln(a*x+1)*x^2*a^2-8*a^2*ln(x)*x^2- 6*a*x+1)/(-a^2*x^2+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\left [\frac {4 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \log \left (\frac {4 \, a^{5} c x^{5} + {\left (2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right )} c x^{6} + {\left (4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right )} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x - {\left (4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left (4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right )} x^{5} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a - 1\right )} x^{2} - 6 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{2} x^{3} - x\right )}}, -\frac {8 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {c} \arctan \left (-\frac {{\left (2 \, a^{2} x^{2} + {\left (2 \, a^{3} + 2 \, a^{2} + a\right )} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} + a^{2}\right )} c x^{4} + {\left (a^{2} + 2 \, a + 1\right )} c x^{2} - 2 \, a c x - c}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a - 1\right )} x^{2} - 6 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{2} x^{3} - x\right )}}\right ] \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorith m="fricas")
Output:
[1/2*(4*(a^3*x^3 - a*x)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c* x^2 - 4*a*c*x - (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5 + 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^ 2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x), -1/ 2*(8*(a^3*x^3 - a*x)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c *x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) - sqrt( -a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))) /(a^2*x^3 - x)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{x^{2} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x) ))/(x**2*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorith m="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*sqrt(c - c/(a^2*x^2))/((a*x + 1)^3*x^2), x)
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.32 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=-\frac {1}{2} \, {\left (8 \, \log \left ({\left | a x + 1 \right |}\right ) \mathrm {sgn}\left (x\right ) - 8 \, \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {6 \, a x \mathrm {sgn}\left (x\right ) - \mathrm {sgn}\left (x\right )}{a^{2} x^{2}}\right )} \sqrt {-c} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorith m="giac")
Output:
-1/2*(8*log(abs(a*x + 1))*sgn(x) - 8*log(abs(x))*sgn(x) - (6*a*x*sgn(x) - sgn(x))/(a^2*x^2))*sqrt(-c)*abs(a)
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^2\,{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2))/(x^2*(a*x + 1)^3),x)
Output:
int(((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2))/(x^2*(a*x + 1)^3), x)
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.28 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx=\frac {\sqrt {c}\, i \left (8 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-8 \,\mathrm {log}\left (x \right ) a^{2} x^{2}-6 a x +1\right )}{2 a \,x^{2}} \] Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x)
Output:
(sqrt(c)*i*(8*log(a*x + 1)*a**2*x**2 - 8*log(x)*a**2*x**2 - 6*a*x + 1))/(2 *a*x**2)