Integrand size = 27, antiderivative size = 113 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {8 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{5 \sqrt {c-\frac {c}{a x}}}+\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c} \] Output:
8/5*a^2*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)+2/5*a^2*(1-1/a^2/x^2)^(1/2)* (c-c/a/x)^(1/2)+2/5*a^2*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/c
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (1-3 a x+6 a^2 x^2\right )}{5 x (-1+a x)} \] Input:
Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^3),x]
Output:
(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(1 - 3*a*x + 6*a^2*x^2))/(5*x *(-1 + a*x))
Time = 0.65 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6733, 572, 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.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-\coth ^{-1}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6733 |
\(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 572 |
\(\displaystyle -\frac {-\frac {3}{5} a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{c}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle -\frac {-\frac {3}{5} a \left (\frac {4}{3} c \int \frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2}{3} a c \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}\right )-\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{c}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle -\frac {-\frac {3}{5} a \left (\frac {8 a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}+\frac {2}{3} a c \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}\right )-\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{c}\) |
Input:
Int[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^3),x]
Output:
-(((-3*a*((8*a*c^2*Sqrt[1 - 1/(a^2*x^2)])/(3*Sqrt[c - c/(a*x)]) + (2*a*c*S qrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/3))/5 - (2*a^2*Sqrt[1 - 1/(a^2*x^2 )]*(c - c/(a*x))^(3/2))/5)/c)
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 = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.53
method | result | size |
orering | \(\frac {2 \left (6 a^{2} x^{2}-3 a x +1\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{5 x^{2} \left (a x -1\right )}\) | \(60\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (6 a^{2} x^{2}-3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{5 x^{2} \left (a x -1\right )}\) | \(62\) |
default | \(\frac {2 \left (a x +1\right ) \left (6 a^{2} x^{2}-3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{5 x^{2} \left (a x -1\right )}\) | \(62\) |
risch | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 a^{3} x^{3}+3 a^{2} x^{2}-2 a x +1\right )}{5 \left (a x -1\right ) x^{2}}\) | \(65\) |
Input:
int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
2/5*(6*a^2*x^2-3*a*x+1)*(a*x+1)/x^2/(a*x-1)*(c-c/a/x)^(1/2)*((a*x-1)/(a*x+ 1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \, {\left (6 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a x^{3} - x^{2}\right )}} \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="fricas ")
Output:
2/5*(6*a^3*x^3 + 3*a^2*x^2 - 2*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a* c*x - c)/(a*x))/(a*x^3 - x^2)
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\text {Timed out} \] Input:
integrate((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(1/2)/x**3,x)
Output:
Timed out
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="maxima ")
Output:
integrate(sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1))/x^3, x)
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^3,x, algorithm="giac")
Output:
integrate(sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1))/x^3, x)
Time = 13.69 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (6\,a^3\,x^3+3\,a^2\,x^2-2\,a\,x+1\right )}{5\,x^2\,\left (a\,x-1\right )} \] Input:
int(((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/x^3,x)
Output:
(2*(c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(3*a^2*x^2 - 2*a*x + 6* a^3*x^3 + 1))/(5*x^2*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \sqrt {c}\, \left (6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-6 a^{3} x^{3}\right )}{5 a \,x^{3}} \] Input:
int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/x^3,x)
Output:
(2*sqrt(c)*(6*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 3*sqrt(x)*sqrt(a)* sqrt(a*x + 1)*a*x + sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 6*a**3*x**3))/(5*a*x** 3)