Integrand size = 27, antiderivative size = 106 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (1+a x)}{\sqrt {1-a^2 x^2}} \] Output:
-(c-c/a^2/x^2)^(1/2)/(-a^2*x^2+1)^(1/2)-3*a*(c-c/a^2/x^2)^(1/2)*x*ln(x)/(- a^2*x^2+1)^(1/2)+4*a*(c-c/a^2/x^2)^(1/2)*x*ln(a*x+1)/(-a^2*x^2+1)^(1/2)
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} (-1-3 a x \log (x)+4 a x \log (1+a x))}{\sqrt {1-a^2 x^2}} \] Input:
Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*(-1 - 3*a*x*Log[x] + 4*a*x*Log[1 + a*x]))/Sqrt[1 - a^2*x^2]
Time = 0.50 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.48, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {1-a^2 x^2}}{x^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x^2 (a x+1)}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \left (\frac {4 a^2}{a x+1}-\frac {3 a}{x}+\frac {1}{x^2}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-3 a \log (x)+4 a \log (a x+1)-\frac {1}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-x^(-1) - 3*a*Log[x] + 4*a*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 = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (4 \ln \left (a x +1\right ) x a -3 a \ln \left (x \right ) x -1\right )}{\sqrt {-a^{2} x^{2}+1}}\) | \(51\) |
Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x,method=_RETURNVER BOSE)
Output:
(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(4*ln(a*x+1)*x*a-3*a*ln(x)*x-1)/(-a^2*x^2+1) ^(1/2)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{3} x} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm= "fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^ 2*x^3 + 2*a*x^2 + x), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{x \left (a x + 1\right )^{3}}\, dx \] Input:
integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x) ))/(x*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{3} x} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm= "maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*sqrt(c - c/(a^2*x^2))/((a*x + 1)^3*x), x)
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\sqrt {-c} {\left (\frac {4 \, \log \left ({\left | a x + 1 \right |}\right ) \mathrm {sgn}\left (x\right )}{a} - \frac {3 \, \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )}{a} - \frac {\mathrm {sgn}\left (x\right )}{a^{2} x}\right )} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm= "giac")
Output:
sqrt(-c)*(4*log(abs(a*x + 1))*sgn(x)/a - 3*log(abs(x))*sgn(x)/a - sgn(x)/( a^2*x))*abs(a)
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x\,{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2))/(x*(a*x + 1)^3),x)
Output:
int(((c - c/(a^2*x^2))^(1/2)*(1 - a^2*x^2)^(3/2))/(x*(a*x + 1)^3), x)
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.26 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {c}\, i \left (-4 \,\mathrm {log}\left (a x +1\right ) a x +3 \,\mathrm {log}\left (x \right ) a x +1\right )}{a x} \] Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x)
Output:
(sqrt(c)*i*( - 4*log(a*x + 1)*a*x + 3*log(x)*a*x + 1))/(a*x)