Integrand size = 27, antiderivative size = 163 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Output:
-4*a^4*(c-c/a/x)^(1/2)-2/3*a^4*(c-c/a/x)^(3/2)/c-2/5*a^4*(c-c/a/x)^(5/2)/c ^2+2/7*a^4*(c-c/a/x)^(7/2)/c^3-2/9*a^4*(c-c/a/x)^(9/2)/c^4+4*2^(1/2)*a^4*c ^(1/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}} \left (35-95 a x+138 a^2 x^2-236 a^3 x^3+788 a^4 x^4\right )}{315 x^4}+4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \] Input:
Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^5),x]
Output:
(-2*Sqrt[c - c/(a*x)]*(35 - 95*a*x + 138*a^2*x^2 - 236*a^3*x^3 + 788*a^4*x ^4))/(315*x^4) + 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]* Sqrt[c])]
Time = 0.66 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {(1-a x) \sqrt {c-\frac {c}{a x}}}{x^5 (a x+1)}dx\) |
| \(\Big \downarrow \) 1070 |
| \(\displaystyle \int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}}}{x^5 \left (a+\frac {1}{x}\right )}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^5}dx}{c}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^3}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a \int \left (\frac {a^2 \left (c-\frac {c}{a x}\right )^{7/2}}{c^2}-\frac {a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{c}+a^2 \left (c-\frac {c}{a x}\right )^{3/2}-\frac {a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{a+\frac {1}{x}}\right )d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (4 \sqrt {2} a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^3}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^2}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c}-\frac {2}{3} a^3 \left (c-\frac {c}{a x}\right )^{3/2}-4 a^3 c \sqrt {c-\frac {c}{a x}}\right )}{c}\) |
Input:
Int[Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^5),x]
Output:
(a*(-4*a^3*c*Sqrt[c - c/(a*x)] - (2*a^3*(c - c/(a*x))^(3/2))/3 - (2*a^3*(c - c/(a*x))^(5/2))/(5*c) + (2*a^3*(c - c/(a*x))^(7/2))/(7*c^2) - (2*a^3*(c - c/(a*x))^(9/2))/(9*c^3) + 4*Sqrt[2]*a^3*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 = 0.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(-\frac {2 \left (788 a^{5} x^{5}-1024 a^{4} x^{4}+374 a^{3} x^{3}-233 a^{2} x^{2}+130 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \left (a x -1\right )}-\frac {2 a^{4} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {c}\, \left (a x -1\right )}\) | \(168\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-1890 \sqrt {a \,x^{2}-x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}+630 a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, x^{6}+1260 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{4}+945 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-630 a^{\frac {9}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{6}-945 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-316 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{3}+156 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{2}-120 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x +70 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{315 x^{5} \sqrt {x \left (a x -1\right )}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(326\) |
Input:
int((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x,method=_RETURNVERBOSE)
Output:
-2/315*(788*a^5*x^5-1024*a^4*x^4+374*a^3*x^3-233*a^2*x^2+130*a*x-35)/x^4*( c*(a*x-1)/a/x)^(1/2)/(a*x-1)-2*a^4*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 = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}, -\frac {2 \, {\left (630 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}\right ] \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="fricas" )
Output:
[2/315*(315*sqrt(2)*a^4*sqrt(c)*x^4*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c* x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - (788*a^4*x^4 - 236*a^3*x^3 + 138 *a^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4, -2/315*(630*sqrt(2)* a^4*sqrt(-c)*x^4*arctan(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c* x - c)) + (788*a^4*x^4 - 236*a^3*x^3 + 138*a^2*x^2 - 95*a*x + 35)*sqrt((a* c*x - c)/(a*x)))/x^4]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=- \int \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x^{6} + x^{5}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{6} + x^{5}}\, dx \] Input:
integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)
Output:
-Integral(-sqrt(c - c/(a*x))/(a*x**6 + x**5), x) - Integral(a*x*sqrt(c - c /(a*x))/(a*x**6 + x**5), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{2} x^{5}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="maxima" )
Output:
-integrate((a^2*x^2 - 1)*sqrt(c - c/(a*x))/((a*x + 1)^2*x^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (136) = 272\).
Time = 0.33 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.66 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {4 \, \sqrt {2} a^{5} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a^{9} c - 1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{8} c^{\frac {3}{2}} {\left | a \right |} + 2100 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{9} c^{2} - 3150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{8} c^{\frac {5}{2}} {\left | a \right |} + 3528 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{9} c^{3} - 2625 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{8} c^{\frac {7}{2}} {\left | a \right |} + 1215 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{9} c^{4} - 315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{8} c^{\frac {9}{2}} {\left | a \right |} + 35 \, a^{9} c^{5}\right )}}{315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right )} \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm="giac")
Output:
4*sqrt(2)*a^5*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)) - 2/315*(1 260*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^8*a^9*c - 1260*(sqrt(a^2*c)* x - sqrt(a^2*c*x^2 - a*c*x))^7*a^8*c^(3/2)*abs(a) + 2100*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a^9*c^2 - 3150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*a^8*c^(5/2)*abs(a) + 3528*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a* c*x))^4*a^9*c^3 - 2625*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^3*a^8*c^( 7/2)*abs(a) + 1215*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a^9*c^4 - 3 15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a^8*c^(9/2)*abs(a) + 35*a^9*c ^5)/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^9*a^4*abs(a)*sgn(x))
Time = 15.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{7\,c^3}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{5\,c^2}-4\,a^4\,\sqrt {c-\frac {c}{a\,x}}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{9/2}}{9\,c^4}-\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \] Input:
int(-((c - c/(a*x))^(1/2)*(a^2*x^2 - 1))/(x^5*(a*x + 1)^2),x)
Output:
(2*a^4*(c - c/(a*x))^(7/2))/(7*c^3) - (2*a^4*(c - c/(a*x))^(3/2))/(3*c) - (2*a^4*(c - c/(a*x))^(5/2))/(5*c^2) - 4*a^4*(c - c/(a*x))^(1/2) - (2*a^4*( c - c/(a*x))^(9/2))/(9*c^4) - 2^(1/2)*a^4*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 = 191, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (-788 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{4} x^{4}+236 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}-138 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}+95 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -35 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{5} x^{5}+315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{5} x^{5}-315 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{5} x^{5}+508 a^{5} x^{5}\right )}{315 a \,x^{5}} \] Input:
int((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x)
Output:
(2*sqrt(c)*( - 788*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**4*x**4 + 236*sqrt(x)*s qrt(a)*sqrt(a*x - 1)*a**3*x**3 - 138*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**2*x* *2 + 95*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x - 35*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 315*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt(2)*i + i)*a**5 *x**5 + 315*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a **5*x**5 - 315*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2 *a*x + 2)*a**5*x**5 + 508*a**5*x**5))/(315*a*x**5)