Integrand size = 27, antiderivative size = 69 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-4 a^2 \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2} \] Output:
-4*a^2*(c-c/a/x)^(1/2)+2*a^2*(c-c/a/x)^(3/2)/c-2/5*a^2*(c-c/a/x)^(5/2)/c^2
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}} \left (1+3 a x+6 a^2 x^2\right )}{5 x^2} \] Input:
Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^3,x]
Output:
(-2*Sqrt[c - c/(a*x)]*(1 + 3*a*x + 6*a^2*x^2))/(5*x^2)
Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6683, 1070, 281, 948, 86, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx\) |
\(\Big \downarrow \) 6683 |
\(\displaystyle \int \frac {(a x+1) \sqrt {c-\frac {c}{a x}}}{x^3 (1-a x)}dx\) |
\(\Big \downarrow \) 1070 |
\(\displaystyle \int \frac {\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}{x^3 \left (\frac {1}{x}-a\right )}dx\) |
\(\Big \downarrow \) 281 |
\(\displaystyle -\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^3}dx}{a}\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x}d\frac {1}{x}}{a}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {c \int \left (\frac {\left (c-\frac {c}{a x}\right )^{3/2} a^2}{c^2}-\frac {3 \sqrt {c-\frac {c}{a x}} a^2}{c}+\frac {2 a^2}{\sqrt {c-\frac {c}{a x}}}\right )d\frac {1}{x}}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c \left (-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^3}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{c^2}-\frac {4 a^3 \sqrt {c-\frac {c}{a x}}}{c}\right )}{a}\) |
Input:
Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^3,x]
Output:
(c*((-4*a^3*Sqrt[c - c/(a*x)])/c + (2*a^3*(c - c/(a*x))^(3/2))/c^2 - (2*a^ 3*(c - c/(a*x))^(5/2))/(5*c^3)))/a
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^ (p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(m + n*(p + r))*( b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 a^{2} x^{2}+3 a x +1\right )}{5 x^{2}}\) | \(35\) |
trager | \(-\frac {2 \left (6 a^{2} x^{2}+3 a x +1\right ) \sqrt {-\frac {-a c x +c}{a x}}}{5 x^{2}}\) | \(37\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 a^{3} x^{3}-3 a^{2} x^{2}-2 a x -1\right )}{5 \left (a x -1\right ) x^{2}}\) | \(50\) |
orering | \(\frac {2 \left (6 a^{2} x^{2}+3 a x +1\right ) \left (a x -1\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}}{5 x^{2} \left (-a^{2} x^{2}+1\right )}\) | \(55\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (10 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x^{4}+10 a^{\frac {7}{2}} \sqrt {x \left (a x -1\right )}\, x^{4}-20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+5 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-5 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-8 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{5 x^{3} \sqrt {x \left (a x -1\right )}\, \sqrt {a}}\) | \(192\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-2/5*(c*(a*x-1)/a/x)^(1/2)*(6*a^2*x^2+3*a*x+1)/x^2
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 \, {\left (6 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, x^{2}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^3,x, algorithm="fricas" )
Output:
-2/5*(6*a^2*x^2 + 3*a*x + 1)*sqrt((a*c*x - c)/(a*x))/x^2
\[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=- \int \frac {\sqrt {c - \frac {c}{a x}}}{a x^{4} - x^{3}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{4} - x^{3}}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**3,x)
Output:
-Integral(sqrt(c - c/(a*x))/(a*x**4 - x**3), x) - Integral(a*x*sqrt(c - c/ (a*x))/(a*x**4 - x**3), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{{\left (a^{2} x^{2} - 1\right )} x^{3}} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^3,x, algorithm="maxima" )
Output:
-integrate((a*x + 1)^2*sqrt(c - c/(a*x))/((a^2*x^2 - 1)*x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (61) = 122\).
Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.38 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 \, {\left (5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{2} c^{\frac {3}{2}} {\left | a \right |} + 5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{3} c^{2} - 5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{2} c^{\frac {5}{2}} {\left | a \right |} + a^{3} c^{3}\right )}}{5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} {\left | a \right |} \mathrm {sgn}\left (x\right )} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^3,x, algorithm="giac")
Output:
-2/5*(5*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^3*a^2*c^(3/2)*abs(a) + 5 *(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a^3*c^2 - 5*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a^2*c^(5/2)*abs(a) + a^3*c^3)/((sqrt(a^2*c)*x - s qrt(a^2*c*x^2 - a*c*x))^5*abs(a)*sgn(x))
Time = 14.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.46 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\left (6\,a^2\,x^2+3\,a\,x+1\right )}{5\,x^2} \] Input:
int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(x^3*(a^2*x^2 - 1)),x)
Output:
-(2*(c - c/(a*x))^(1/2)*(3*a*x + 6*a^2*x^2 + 1))/(5*x^2)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \sqrt {c}\, \left (-6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}-3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -\sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+6 a^{3} x^{3}\right )}{5 a \,x^{3}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(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)