Integrand size = 27, antiderivative size = 152 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=-\frac {\sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \] Output:
-1/2*(-a^2*c*x^2+c)^(1/2)/a/(1-1/a^2/x^2)^(1/2)/x^3+3*(-a^2*c*x^2+c)^(1/2) /(1-1/a^2/x^2)^(1/2)/x^2+4*a*(-a^2*c*x^2+c)^(1/2)*ln(x)/(1-1/a^2/x^2)^(1/2 )/x-4*a*(-a^2*c*x^2+c)^(1/2)*ln(a*x+1)/(1-1/a^2/x^2)^(1/2)/x
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {1}{2 a x^2}+\frac {3}{x}+4 a \log (x)-4 a \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^3),x]
Output:
(Sqrt[c - a^2*c*x^2]*(-1/2*1/(a*x^2) + 3/x + 4*a*Log[x] - 4*a*Log[1 + a*x] ))/(Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.83 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6746, 6747, 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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2} e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}dx}{x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {(1-a x)^2}{x^3 (a x+1)}dx}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \left (-\frac {4 a^3}{a x+1}+\frac {4 a^2}{x}-\frac {3 a}{x^2}+\frac {1}{x^3}\right )dx}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^3),x]
Output:
(Sqrt[c - a^2*c*x^2]*(-1/2*1/x^2 + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)
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^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {\left (8 a^{2} \ln \left (x \right ) x^{2}-8 \ln \left (a x +1\right ) x^{2} a^{2}+6 a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 x^{2} \left (a x -1\right )^{2}}\) | \(77\) |
Input:
int((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x,method=_RETURNVERBO SE)
Output:
1/2*(8*a^2*ln(x)*x^2-8*ln(a*x+1)*x^2*a^2+6*a*x-1)*(-c*(a^2*x^2-1))^(1/2)*( a*x+1)*((a*x-1)/(a*x+1))^(3/2)/x^2/(a*x-1)^2
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {8 \, a^{3} \sqrt {-c} x^{2} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {-a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {-c} + a c}{a x^{2} + x}\right ) + \sqrt {-a^{2} c} {\left (6 \, a x - 1\right )}}{2 \, a x^{2}} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="f ricas")
Output:
1/2*(8*a^3*sqrt(-c)*x^2*log((2*a^3*c*x^2 + 2*a^2*c*x + sqrt(-a^2*c)*(2*a*x + 1)*sqrt(-c) + a*c)/(a*x^2 + x)) + sqrt(-a^2*c)*(6*a*x - 1))/(a*x^2)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\text {Timed out} \] Input:
integrate((-a**2*c*x**2+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**3,x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="m axima")
Output:
integrate(sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(3/2)/x^3, x)
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=-\frac {1}{2} \, {\left (8 \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) \mathrm {sgn}\left (a x + 1\right ) - 8 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + 1\right ) - \frac {6 \, a x \mathrm {sgn}\left (a x + 1\right ) - \mathrm {sgn}\left (a x + 1\right )}{x^{2}}\right )} \sqrt {-c} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="g iac")
Output:
-1/2*(8*a^2*log(abs(a*x + 1))*sgn(a*x + 1) - 8*a^2*log(abs(x))*sgn(a*x + 1 ) - (6*a*x*sgn(a*x + 1) - sgn(a*x + 1))/x^2)*sqrt(-c)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^3} \,d x \] Input:
int(((c - a^2*c*x^2)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^3,x)
Output:
int(((c - a^2*c*x^2)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^3, x)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int \frac {\sqrt {-a^{2} c \,x^{2}+c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{x^{3}}d x \] Input:
int((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x)
Output:
int((-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x)