Integrand size = 27, antiderivative size = 150 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {224 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{15 \sqrt {c-\frac {c}{a x}}}-\frac {56}{15} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}-\frac {7 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}-\frac {a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}} \] Output:
-224/15*a^2*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)-56/15*a^2*(1-1/a^2/x^2)^ (1/2)*(c-c/a/x)^(1/2)-7/5*a^2*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/c-a^2*(c -c/a/x)^(7/2)/c^3/(1-1/a^2/x^2)^(1/2)
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (3-16 a x+79 a^2 x^2+158 a^3 x^3\right )}{15 x \left (-1+a^2 x^2\right )} \] Input:
Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^3),x]
Output:
(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 - 16*a*x + 79*a^2*x^2 + 1 58*a^3*x^3))/(15*x*(-1 + a^2*x^2))
Time = 0.72 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6733, 572, 459, 459, 458}
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.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6733 |
\(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{7/2}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}d\frac {1}{x}}{c^3}\) |
\(\Big \downarrow \) 572 |
\(\displaystyle -\frac {-\frac {7}{5} a \int \frac {\left (c-\frac {c}{a x}\right )^{7/2}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle -\frac {-\frac {7}{5} a \left (\frac {8}{3} c \int \frac {\left (c-\frac {c}{a x}\right )^{5/2}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {2 a c \left (c-\frac {c}{a x}\right )^{5/2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}\right )-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle -\frac {-\frac {7}{5} a \left (\frac {8}{3} c \left (4 c \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {2 a c \left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 a c \left (c-\frac {c}{a x}\right )^{5/2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}\right )-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle -\frac {-\frac {7}{5} a \left (\frac {8}{3} c \left (\frac {2 a c \left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 a c^2 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 a c \left (c-\frac {c}{a x}\right )^{5/2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}\right )-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
Input:
Int[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^3),x]
Output:
-(((-7*a*((8*c*((-8*a*c^2*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a^2*x^2)] + (2*a* c*(c - c/(a*x))^(3/2))/Sqrt[1 - 1/(a^2*x^2)]))/3 + (2*a*c*(c - c/(a*x))^(5 /2))/(3*Sqrt[1 - 1/(a^2*x^2)])))/5 - (2*a^2*(c - c/(a*x))^(7/2))/(5*Sqrt[1 - 1/(a^2*x^2)]))/c^3)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] + Simp[c*(n/(d *(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && NeQ[n + 2*p + 2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_S ymbol] :> Simp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && Int egerQ[(n - 1)/2] && IntegerQ[m] && IntegerQ[2*p]
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.45
method | result | size |
orering | \(-\frac {2 \left (158 a^{3} x^{3}+79 a^{2} x^{2}-16 a x +3\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{15 x^{2} \left (a x -1\right )^{2}}\) | \(68\) |
gosper | \(-\frac {2 \left (a x +1\right ) \left (158 a^{3} x^{3}+79 a^{2} x^{2}-16 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{15 x^{2} \left (a x -1\right )^{2}}\) | \(70\) |
default | \(-\frac {2 \left (a x +1\right ) \left (158 a^{3} x^{3}+79 a^{2} x^{2}-16 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{15 x^{2} \left (a x -1\right )^{2}}\) | \(70\) |
risch | \(-\frac {2 \left (98 a^{3} x^{3}+79 a^{2} x^{2}-16 a x +3\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} \left (a x -1\right )}-\frac {8 a^{3} x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\) | \(109\) |
Input:
int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-2/15*(158*a^3*x^3+79*a^2*x^2-16*a*x+3)*(a*x+1)/x^2/(a*x-1)^2*(c-c/a/x)^(1 /2)*((a*x-1)/(a*x+1))^(3/2)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 \, {\left (158 \, a^{3} x^{3} + 79 \, a^{2} x^{2} - 16 \, a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{15 \, {\left (a x^{3} - x^{2}\right )}} \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="fricas ")
Output:
-2/15*(158*a^3*x^3 + 79*a^2*x^2 - 16*a*x + 3)*sqrt((a*x - 1)/(a*x + 1))*sq rt((a*c*x - c)/(a*x))/(a*x^3 - x^2)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\text {Timed out} \] Input:
integrate((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**3,x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="maxima ")
Output:
integrate(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)/x^3, x)
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x, algorithm="giac")
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
Time = 13.65 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (158\,a^3\,x^3+79\,a^2\,x^2-16\,a\,x+3\right )}{15\,x^2\,\left (a\,x-1\right )} \] Input:
int(((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^3,x)
Output:
-(2*(c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(79*a^2*x^2 - 16*a*x + 158*a^3*x^3 + 3))/(15*x^2*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \sqrt {c}\, \left (158 \sqrt {a x +1}\, a^{3} x^{3}-158 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}-79 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+16 \sqrt {x}\, \sqrt {a}\, a x -3 \sqrt {x}\, \sqrt {a}\right )}{15 \sqrt {a x +1}\, a \,x^{3}} \] Input:
int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^3,x)
Output:
(2*sqrt(c)*(158*sqrt(a*x + 1)*a**3*x**3 - 158*sqrt(x)*sqrt(a)*a**3*x**3 - 79*sqrt(x)*sqrt(a)*a**2*x**2 + 16*sqrt(x)*sqrt(a)*a*x - 3*sqrt(x)*sqrt(a)) )/(15*sqrt(a*x + 1)*a*x**3)