Integrand size = 23, antiderivative size = 148 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}+\frac {11 a c \sqrt {1-a^2 x^2}}{12 x^2 \sqrt {c-a c x}}-\frac {11 a^2 c \sqrt {1-a^2 x^2}}{8 x \sqrt {c-a c x}}+\frac {11}{8} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
11/8*a^3*arctanh(c^(1/2)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2))*c^(1/2)-1/3* c*(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c)^(1/2)+11/12*a*c*(-a^2*x^2+1)^(1/2)/x^2 /(-a*c*x+c)^(1/2)-11/8*a^2*c*(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.38 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {c \sqrt {1-a^2 x^2} \left (-1+11 a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1+a x\right )\right )}{3 x^3 \sqrt {c-a c x}} \]
(c*Sqrt[1 - a^2*x^2]*(-1 + 11*a^3*x^3*Hypergeometric2F1[1/2, 3, 3/2, 1 + a *x]))/(3*x^3*Sqrt[c - a*c*x])
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6678, 580, 579, 579, 573, 219}
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.
| \(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{x^4 \sqrt {1-a^2 x^2}}dx}{c}\) |
| \(\Big \downarrow \) 580 |
| \(\displaystyle \frac {-\frac {11}{6} a c \int \frac {\sqrt {c-a c x}}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {c^2 \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}}{c}\) |
| \(\Big \downarrow \) 579 |
| \(\displaystyle \frac {-\frac {11}{6} a c \left (-\frac {3}{4} a \int \frac {\sqrt {c-a c x}}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}}{c}\) |
| \(\Big \downarrow \) 579 |
| \(\displaystyle \frac {-\frac {11}{6} a c \left (-\frac {3}{4} a \left (-\frac {1}{2} a \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}}dx-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}}{c}\) |
| \(\Big \downarrow \) 573 |
| \(\displaystyle \frac {-\frac {11}{6} a c \left (-\frac {3}{4} a \left (a c \int \frac {1}{1-\frac {c \left (1-a^2 x^2\right )}{c-a c x}}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}}{c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {11}{6} a c \left (-\frac {3}{4} a \left (a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}}{c}\) |
(-1/3*(c^2*Sqrt[1 - a^2*x^2])/(x^3*Sqrt[c - a*c*x]) - (11*a*c*(-1/2*(c*Sqr t[1 - a^2*x^2])/(x^2*Sqrt[c - a*c*x]) - (3*a*(-((c*Sqrt[1 - a^2*x^2])/(x*S qrt[c - a*c*x])) + a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]))/4))/6)/c
3.5.18.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*c*e*(m + 1))), x] - Simp[d*((n - m - 2)/(c*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && LtQ[m, -1] && (IntegerQ[2*p] | | IntegerQ[m])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ [p + 1/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (33 c \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3}-33 a^{2} x^{2} \sqrt {c \left (a x +1\right )}\, \sqrt {c}+22 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}-8 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{24 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{3}}\) | \(121\) |
| risch | \(\frac {\left (33 a^{3} x^{3}+11 a^{2} x^{2}-14 a x +8\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{24 x^{3} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {11 a^{3} \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(158\) |
-1/24*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(33*c*arctanh((c*(a*x+1))^(1/2 )/c^(1/2))*a^3*x^3-33*a^2*x^2*(c*(a*x+1))^(1/2)*c^(1/2)+22*a*x*(c*(a*x+1)) ^(1/2)*c^(1/2)-8*(c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/(c*(a*x+1))^(1 /2)/x^3
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {33 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (33 \, a^{2} x^{2} - 22 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {33 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (33 \, a^{2} x^{2} - 22 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
[1/48*(33*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^ 2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(33*a^2*x^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24*(33*( a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqr t(-c)/(a^2*c*x^2 - c)) + (33*a^2*x^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt (-a*c*x + c))/(a*x^4 - x^3)]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{{\left (a x + 1\right )} x^{4}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {1}{24} \, a^{3} c^{2} {\left (\frac {33 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{2}} + \frac {33 \, {\left (a c x + c\right )}^{\frac {5}{2}} - 88 \, {\left (a c x + c\right )}^{\frac {3}{2}} c + 63 \, \sqrt {a c x + c} c^{2}}{a^{3} c^{5} x^{3}}\right )} {\left | c \right |} + \frac {33 \, a^{3} c {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + 19 \, \sqrt {2} a^{3} \sqrt {-c} \sqrt {c} {\left | c \right |}}{24 \, \sqrt {-c} c} \]
-1/24*a^3*c^2*(33*arctan(sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c^2) + (33*(a *c*x + c)^(5/2) - 88*(a*c*x + c)^(3/2)*c + 63*sqrt(a*c*x + c)*c^2)/(a^3*c^ 5*x^3))*abs(c) + 1/24*(33*a^3*c*abs(c)*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + 19*sqrt(2)*a^3*sqrt(-c)*sqrt(c)*abs(c))/(sqrt(-c)*c)
Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^4\,\left (a\,x+1\right )} \,d x \]