Integrand size = 27, antiderivative size = 221 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-a^2 c x^2}}{x^3 \sqrt {1-a^2 x^2}}-\frac {2 a^2 \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}+\frac {4 a^4 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^4 \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-a^2 x^2}} \]
-1/4*(-a^2*c*x^2+c)^(1/2)/x^4/(-a^2*x^2+1)^(1/2)+a*(-a^2*c*x^2+c)^(1/2)/x^ 3/(-a^2*x^2+1)^(1/2)-2*a^2*(-a^2*c*x^2+c)^(1/2)/x^2/(-a^2*x^2+1)^(1/2)+4*a ^3*(-a^2*c*x^2+c)^(1/2)/x/(-a^2*x^2+1)^(1/2)+4*a^4*ln(x)*(-a^2*c*x^2+c)^(1 /2)/(-a^2*x^2+1)^(1/2)-4*a^4*ln(a*x+1)*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^( 1/2)
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.35 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {1}{4 x^4}+\frac {a}{x^3}-\frac {2 a^2}{x^2}+\frac {4 a^3}{x}+4 a^4 \log (x)-4 a^4 \log (1+a x)\right )}{\sqrt {1-a^2 x^2}} \]
(Sqrt[c - a^2*c*x^2]*(-1/4*1/x^4 + a/x^3 - (2*a^2)/x^2 + (4*a^3)/x + 4*a^4 *Log[x] - 4*a^4*Log[1 + a*x]))/Sqrt[1 - a^2*x^2]
Time = 0.46 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6703, 6700, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {1-a^2 x^2}}{x^5}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {(1-a x)^2}{x^5 (a x+1)}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \left (-\frac {4 a^5}{a x+1}+\frac {4 a^4}{x}-\frac {4 a^3}{x^2}+\frac {4 a^2}{x^3}-\frac {3 a}{x^4}+\frac {1}{x^5}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (4 a^4 \log (x)-4 a^4 \log (a x+1)+\frac {4 a^3}{x}-\frac {2 a^2}{x^2}+\frac {a}{x^3}-\frac {1}{4 x^4}\right )}{\sqrt {1-a^2 x^2}}\) |
(Sqrt[c - a^2*c*x^2]*(-1/4*1/x^4 + a/x^3 - (2*a^2)/x^2 + (4*a^3)/x + 4*a^4 *Log[x] - 4*a^4*Log[1 + a*x]))/Sqrt[1 - a^2*x^2]
3.13.70.3.1 Defintions of rubi rules used
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.40
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (16 \ln \left (a x +1\right ) x^{4} a^{4}-16 \ln \left (x \right ) x^{4} a^{4}-16 a^{3} x^{3}+8 a^{2} x^{2}-4 a x +1\right )}{4 \left (a^{2} x^{2}-1\right ) x^{4}}\) | \(89\) |
1/4*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(16*ln(a*x+1)*x^4*a^4-16*ln( x)*x^4*a^4-16*a^3*x^3+8*a^2*x^2-4*a*x+1)/(a^2*x^2-1)/x^4
Time = 0.29 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.28 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\left [\frac {8 \, {\left (a^{6} x^{6} - a^{4} x^{4}\right )} \sqrt {c} \log \left (\frac {4 \, a^{5} c x^{5} + {\left (2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right )} c x^{6} + {\left (4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right )} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} x^{4} + 6 \, a^{2} x^{2} + 4 \, a x + 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right ) - {\left (16 \, a^{3} x^{3} - {\left (16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right )} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\left (a^{2} x^{6} - x^{4}\right )}}, \frac {16 \, {\left (a^{6} x^{6} - a^{4} x^{4}\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} + 2 \, a + 1\right )} x^{2} + 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} + a^{2}\right )} c x^{4} + {\left (a^{2} + 2 \, a + 1\right )} c x^{2} - 2 \, a c x - c}\right ) - {\left (16 \, a^{3} x^{3} - {\left (16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right )} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\left (a^{2} x^{6} - x^{4}\right )}}\right ] \]
[1/4*(8*(a^6*x^6 - a^4*x^4)*sqrt(c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6* a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2 *c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 + 6*a^2 + 4*a + 1)*x^4 + 6*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) - (16*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)* x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2 *x^6 - x^4), 1/4*(16*(a^6*x^6 - a^4*x^4)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 + 2*a + 1)*x^2 + 2*a*x + 1)*sqrt(-c)/(2*a^ 3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) - (1 6*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)*x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(- a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2*x^6 - x^4)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{x^{5} \left (a x + 1\right )^{3}}\, dx \]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{5}} \,d x } \]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{5}} \,d x } \]
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^5\,{\left (a\,x+1\right )}^3} \,d x \]