Integrand size = 27, antiderivative size = 107 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (1-a x)}{\sqrt {1-a^2 x^2}} \] 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 = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} (-1+3 a x \log (x)-4 a x \log (1-a x))}{\sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/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 = 0.82 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.49, 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 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(a x+1)^2}{x^2 (1-a x)}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (3 a \log (x)-4 a \log (1-a x)-\frac {1}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/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 = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (4 a \ln \left (a x -1\right ) x -3 a \ln \left (x \right ) x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\) | \(52\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x,x,method=_RETURNVER BOSE)
Output:
-(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(4*a*ln(a*x-1)*x-3*a*ln(x)*x+1)/(-a^2*x^2+1 )^(1/2)
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/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)
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x,x)
Output:
Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x*(-(a*x - 1) *(a*x + 1))**(3/2)), x)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.35 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {1}{2} \, a^{3} {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{3}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{3}}\right )} - \frac {3}{2} \, a^{2} {\left (\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{2}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{2}}\right )} - \frac {3}{2} \, a {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a} + \frac {2 i \, \sqrt {c} \log \left (x\right )}{a}\right )} - \frac {1}{2} i \, \sqrt {c} \log \left (a x + 1\right ) + \frac {1}{2} i \, \sqrt {c} \log \left (a x - 1\right ) + \frac {i \, \sqrt {c}}{a x} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x,x, algorithm= "maxima")
Output:
-1/2*a^3*(-I*sqrt(c)*log(a*x + 1)/a^3 - I*sqrt(c)*log(a*x - 1)/a^3) - 3/2* a^2*(I*sqrt(c)*log(a*x + 1)/a^2 - I*sqrt(c)*log(a*x - 1)/a^2) - 3/2*a*(-I* sqrt(c)*log(a*x + 1)/a - I*sqrt(c)*log(a*x - 1)/a + 2*I*sqrt(c)*log(x)/a) - 1/2*I*sqrt(c)*log(a*x + 1) + 1/2*I*sqrt(c)*log(a*x - 1) + I*sqrt(c)/(a*x )
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.36 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {{\left (4 \, a \log \left ({\left | a x - 1 \right |}\right ) \mathrm {sgn}\left (x\right ) - 3 \, a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {\mathrm {sgn}\left (x\right )}{x}\right )} \sqrt {-c} {\left | a \right |}}{a^{2}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x,x, algorithm= "giac")
Output:
-(4*a*log(abs(a*x - 1))*sgn(x) - 3*a*log(abs(x))*sgn(x) + sgn(x)/x)*sqrt(- c)*abs(a)/a^2
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^3)/(x*(1 - a^2*x^2)^(3/2)),x)
Output:
int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^3)/(x*(1 - a^2*x^2)^(3/2)), 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((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x,x)
Output:
(sqrt(c)*i*( - 4*log(a*x - 1)*a*x + 3*log(x)*a*x - 1))/(a*x)