Integrand size = 23, antiderivative size = 112 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
-7/4*a^2*arctanh(c^(1/2)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2))*c^(1/2)-1/2* c*(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^(1/2)+7/4*a*c*(-a^2*x^2+1)^(1/2)/x/(-a *c*x+c)^(1/2)
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=-\frac {c \sqrt {1-a x} \left ((2-7 a x) \sqrt {1+a x}+7 a^2 x^2 \text {arctanh}\left (\sqrt {1+a x}\right )\right )}{4 x^2 \sqrt {c-a c x}} \]
-1/4*(c*Sqrt[1 - a*x]*((2 - 7*a*x)*Sqrt[1 + a*x] + 7*a^2*x^2*ArcTanh[Sqrt[ 1 + a*x]]))/(x^2*Sqrt[c - a*c*x])
Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6678, 580, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{x^3 \sqrt {1-a^2 x^2}}dx}{c}\) |
\(\Big \downarrow \) 580 |
\(\displaystyle \frac {-\frac {7}{4} a c \int \frac {\sqrt {c-a c x}}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 579 |
\(\displaystyle \frac {-\frac {7}{4} a c \left (-\frac {1}{2} a \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}}dx-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 573 |
\(\displaystyle \frac {-\frac {7}{4} a c \left (a c \int \frac {1}{1-\frac {c \left (1-a^2 x^2\right )}{c-a c x}}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {-\frac {7}{4} a c \left (a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}}{c}\) |
(-1/2*(c^2*Sqrt[1 - a^2*x^2])/(x^2*Sqrt[c - a*c*x]) - (7*a*c*(-((c*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a*c*x])) + a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2 *x^2])/Sqrt[c - a*c*x]]))/4)/c
3.5.17.3.1 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 = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 c \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{2} x^{2}-7 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}+2 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{2}}\) | \(101\) |
risch | \(-\frac {\left (7 a^{2} x^{2}+5 a x -2\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{4 x^{2} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {7 a^{2} \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(150\) |
1/4*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(7*c*arctanh((c*(a*x+1))^(1/2)/c ^(1/2))*a^2*x^2-7*a*x*(c*(a*x+1))^(1/2)*c^(1/2)+2*(c*(a*x+1))^(1/2)*c^(1/2 ))/c^(1/2)/(a*x-1)/(c*(a*x+1))^(1/2)/x^2
Time = 0.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\left [\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{8 \, {\left (a x^{3} - x^{2}\right )}}, -\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{4 \, {\left (a x^{3} - x^{2}\right )}}\right ] \]
[1/8*(7*(a^3*x^3 - a^2*x^2)*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*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2), -1/4*(7*(a^3*x^3 - a^2*x^2 )*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2)]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{{\left (a x + 1\right )} x^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {1}{4} \, a^{2} c {\left (\frac {7 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {7 \, {\left (a c x + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {a c x + c} c}{a^{2} c^{3} x^{2}}\right )} {\left | c \right |} - \frac {7 \, a^{2} c {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + 5 \, \sqrt {2} a^{2} \sqrt {-c} \sqrt {c} {\left | c \right |}}{4 \, \sqrt {-c} c} \]
1/4*a^2*c*(7*arctan(sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c) + (7*(a*c*x + c )^(3/2) - 9*sqrt(a*c*x + c)*c)/(a^2*c^3*x^2))*abs(c) - 1/4*(7*a^2*c*abs(c) *arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + 5*sqrt(2)*a^2*sqrt(-c)*sqrt(c)*abs(c)) /(sqrt(-c)*c)
Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^3\,\left (a\,x+1\right )} \,d x \]