Integrand size = 27, antiderivative size = 113 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=4 a^3 \sqrt {c-\frac {c}{a x}}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Output:
4*a^3*(c-c/a/x)^(1/2)+2/3*a^3*(c-c/a/x)^(3/2)/c+2/7*a^3*(c-c/a/x)^(7/2)/c^ 3-4*2^(1/2)*a^3*c^(1/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-3+9 a x-16 a^2 x^2+52 a^3 x^3\right )}{21 x^3}-4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Input:
Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^4),x]
Output:
(2*Sqrt[c - c/(a*x)]*(-3 + 9*a*x - 16*a^2*x^2 + 52*a^3*x^3))/(21*x^3) - 4* Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]
Time = 0.97 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03, 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, 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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {(1-a x) \sqrt {c-\frac {c}{a x}}}{x^4 (a x+1)}dx\) |
| \(\Big \downarrow \) 1070 |
| \(\displaystyle \int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}}}{x^4 \left (a+\frac {1}{x}\right )}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^4}dx}{c}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^2}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a \int \left (\frac {a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a+\frac {1}{x}}-\frac {a \left (c-\frac {c}{a x}\right )^{5/2}}{c}\right )d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (-4 \sqrt {2} a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^2}+\frac {2}{3} a^2 \left (c-\frac {c}{a x}\right )^{3/2}+4 a^2 c \sqrt {c-\frac {c}{a x}}\right )}{c}\) |
Input:
Int[Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^4),x]
Output:
(a*(4*a^2*c*Sqrt[c - c/(a*x)] + (2*a^2*(c - c/(a*x))^(3/2))/3 + (2*a^2*(c - c/(a*x))^(7/2))/(7*c^2) - 4*Sqrt[2]*a^2*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x) ]/(Sqrt[2]*Sqrt[c])]))/c
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[(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 = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {2 \left (52 a^{4} x^{4}-68 a^{3} x^{3}+25 a^{2} x^{2}-12 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{21 x^{3} \left (a x -1\right )}+\frac {2 a^{3} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {c}\, \left (a x -1\right )}\) | \(160\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (42 \sqrt {x \left (a x -1\right )}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{5}-126 \sqrt {a \,x^{2}-x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{5}+84 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{3}+63 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-42 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{5}-63 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-20 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{2}+12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x -6 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{21 x^{4} \sqrt {x \left (a x -1\right )}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(302\) |
Input:
int((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x,method=_RETURNVERBOSE)
Output:
2/21*(52*a^4*x^4-68*a^3*x^3+25*a^2*x^2-12*a*x+3)/x^3*(c*(a*x-1)/a/x)^(1/2) /(a*x-1)+2*a^3*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2*2^(1/2)*c^(1/2)*((x +1/a)^2*a^2*c-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a))*(c*(a*x-1)*a*x)^(1/2)*(c* (a*x-1)/a/x)^(1/2)/(a*x-1)
Time = 0.07 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} a^{3} \sqrt {c} x^{3} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, x^{3}}, \frac {2 \, {\left (42 \, \sqrt {2} a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, x^{3}}\right ] \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="fricas" )
Output:
[2/21*(21*sqrt(2)*a^3*sqrt(c)*x^3*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + (52*a^3*x^3 - 16*a^2*x^2 + 9*a*x - 3)*sqrt((a*c*x - c)/(a*x)))/x^3, 2/21*(42*sqrt(2)*a^3*sqrt(-c)*x^3*arctan( sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + (52*a^3*x^3 - 16*a^2*x^2 + 9*a*x - 3)*sqrt((a*c*x - c)/(a*x)))/x^3]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=- \int \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x^{5} + x^{4}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{5} + x^{4}}\, dx \] Input:
integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)
Output:
-Integral(-sqrt(c - c/(a*x))/(a*x**5 + x**4), x) - Integral(a*x*sqrt(c - c /(a*x))/(a*x**5 + x**4), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{2} x^{4}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="maxima" )
Output:
-integrate((a^2*x^2 - 1)*sqrt(c - c/(a*x))/((a*x + 1)^2*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (94) = 188\).
Time = 0.27 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.15 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {4 \, \sqrt {2} a^{4} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (84 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{7} c - 84 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{6} c^{\frac {3}{2}} {\left | a \right |} + 112 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{7} c^{2} - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{6} c^{\frac {5}{2}} {\left | a \right |} + 63 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{7} c^{3} - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{6} c^{\frac {7}{2}} {\left | a \right |} + 3 \, a^{7} c^{4}\right )}}{21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{3} {\left | a \right |} \mathrm {sgn}\left (x\right )} \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="giac")
Output:
-4*sqrt(2)*a^4*c*arctan(-1/2*sqrt(2)*((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a* c*x))*a + sqrt(c)*abs(a))/(a*sqrt(-c)))/(sqrt(-c)*abs(a)*sgn(x)) + 2/21*(8 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a^7*c - 84*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*a^6*c^(3/2)*abs(a) + 112*(sqrt(a^2*c)*x - sqrt( a^2*c*x^2 - a*c*x))^4*a^7*c^2 - 105*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c* x))^3*a^6*c^(5/2)*abs(a) + 63*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2* a^7*c^3 - 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a^6*c^(7/2)*abs(a) + 3*a^7*c^4)/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^7*a^3*abs(a)*sgn(x ))
Time = 23.97 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=4\,a^3\,\sqrt {c-\frac {c}{a\,x}}+\frac {2\,a^3\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}+\frac {2\,a^3\,{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{7\,c^3}+\sqrt {2}\,a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \] Input:
int(-((c - c/(a*x))^(1/2)*(a^2*x^2 - 1))/(x^4*(a*x + 1)^2),x)
Output:
4*a^3*(c - c/(a*x))^(1/2) + (2*a^3*(c - c/(a*x))^(3/2))/(3*c) + (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3) + 2^(1/2)*a^3*c^(1/2)*atan((2^(1/2)*(c - c/(a*x) )^(1/2)*1i)/(2*c^(1/2)))*4i
Time = 0.17 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c}\, \left (52 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}-16 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}+9 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{4} x^{4}-21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{4} x^{4}+21 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{4} x^{4}-28 a^{4} x^{4}\right )}{21 a \,x^{4}} \] Input:
int((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x)
Output:
(2*sqrt(c)*(52*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**3*x**3 - 16*sqrt(x)*sqrt(a )*sqrt(a*x - 1)*a**2*x**2 + 9*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x - 3*sqrt(x )*sqrt(a)*sqrt(a*x - 1) - 21*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt(2)*i + i)*a**4*x**4 - 21*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a**4*x**4 + 21*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2)*a**4*x**4 - 28*a**4*x**4))/(21*a*x**4)