Integrand size = 27, antiderivative size = 128 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {12 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{5 \sqrt {1-a x}}+\frac {6 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{5 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{5 x^2 (1-a x)} \]
-12/5*a^2*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a*x+1)^(1/2)+6/5*a*(c-c/a/x)^(1/ 2)*(a*x+1)^(1/2)/x/(-a*x+1)^(1/2)-2/5*(c-c/a/x)^(1/2)*(-a^2*x^2+1)^(1/2)/x ^2/(-a*x+1)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x} \left (1-3 a x+6 a^2 x^2\right )}{5 x^2 \sqrt {1-a x}} \]
Time = 0.48 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6684, 6678, 516, 87, 55, 48}
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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{-\text {arctanh}(a x)} \sqrt {1-a x}}{x^{7/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(1-a x)^{3/2}}{x^{7/2} \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 516 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {1-a x}{x^{7/2} \sqrt {a x+1}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {9}{5} a \int \frac {1}{x^{5/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{5 x^{5/2}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {9}{5} a \left (-\frac {2}{3} a \int \frac {1}{x^{3/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1}}{5 x^{5/2}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\sqrt {x} \left (-\frac {9}{5} a \left (\frac {4 a \sqrt {a x+1}}{3 \sqrt {x}}-\frac {2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1}}{5 x^{5/2}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
(Sqrt[c - c/(a*x)]*Sqrt[x]*((-2*Sqrt[1 + a*x])/(5*x^(5/2)) - (9*a*((-2*Sqr t[1 + a*x])/(3*x^(3/2)) + (4*a*Sqrt[1 + a*x])/(3*Sqrt[x])))/5))/Sqrt[1 - a *x]
3.6.97.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {2 \left (6 a^{2} x^{2}-3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{5 x^{2} \left (a x -1\right )}\) | \(54\) |
default | \(\frac {2 \left (6 a^{2} x^{2}-3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{5 x^{2} \left (a x -1\right )}\) | \(54\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (6 a^{3} x^{3}+3 a^{2} x^{2}-2 a x +1\right )}{5 \sqrt {-a^{2} x^{2}+1}\, x^{2} \sqrt {-\left (a x +1\right ) a c x}}\) | \(90\) |
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \, {\left (6 \, a^{2} x^{2} - 3 \, a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a x^{3} - x^{2}\right )}} \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{3}} \,d x } \]
Time = 3.83 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {2\,\sqrt {1-a^2\,x^2}}{5\,a}-\frac {6\,x\,\sqrt {1-a^2\,x^2}}{5}+\frac {12\,a\,x^2\,\sqrt {1-a^2\,x^2}}{5}\right )}{x^3-\frac {x^2}{a}} \]