Integrand size = 27, antiderivative size = 130 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {7}{8} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
7/8*a^4*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))*c^(1/2)+1/4*(-a^2*c*x^2+c)^( 1/2)/x^4-2/3*a*(-a^2*c*x^2+c)^(1/2)/x^3+7/8*a^2*(-a^2*c*x^2+c)^(1/2)/x^2-4 /3*a^3*(-a^2*c*x^2+c)^(1/2)/x
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (6-16 a x+21 a^2 x^2-32 a^3 x^3\right )}{24 x^4}-\frac {7}{8} a^4 \sqrt {c} \log (x)+\frac {7}{8} a^4 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \]
(Sqrt[c - a^2*c*x^2]*(6 - 16*a*x + 21*a^2*x^2 - 32*a^3*x^3))/(24*x^4) - (7 *a^4*Sqrt[c]*Log[x])/8 + (7*a^4*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2 ]])/8
Time = 0.57 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6717, 6702, 540, 27, 539, 27, 539, 27, 534, 243, 73, 221}
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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2} e^{-2 \coth ^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5}dx\) |
\(\Big \downarrow \) 6702 |
\(\displaystyle -c \int \frac {(1-a x)^2}{x^5 \sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 540 |
\(\displaystyle -c \left (-\frac {\int \frac {a c (8-7 a x)}{x^4 \sqrt {c-a^2 c x^2}}dx}{4 c}-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (-\frac {1}{4} a \int \frac {8-7 a x}{x^4 \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {\int \frac {a c (21-16 a x)}{x^3 \sqrt {c-a^2 c x^2}}dx}{3 c}-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \int \frac {21-16 a x}{x^3 \sqrt {c-a^2 c x^2}}dx-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {\int \frac {a c (32-21 a x)}{x^2 \sqrt {c-a^2 c x^2}}dx}{2 c}-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \int \frac {32-21 a x}{x^2 \sqrt {c-a^2 c x^2}}dx-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (-21 a \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (-\frac {21}{2} a \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (\frac {21 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a c}-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -c \left (-\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (\frac {21 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\) |
-(c*(-1/4*Sqrt[c - a^2*c*x^2]/(c*x^4) - (a*((-8*Sqrt[c - a^2*c*x^2])/(3*c* x^3) - (a*((-21*Sqrt[c - a^2*c*x^2])/(2*c*x^2) - (a*((-32*Sqrt[c - a^2*c*x ^2])/(c*x) + (21*a*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c]))/2))/3)) /4))
3.8.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^(n/2) Int[x^m*((c + d*x^2)^(p + n/2)/(1 - a*x)^n), x] , x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[n/2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {\left (32 a^{5} x^{5}-21 a^{4} x^{4}-16 a^{3} x^{3}+15 a^{2} x^{2}-16 a x +6\right ) c}{24 x^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {7 a^{4} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\) | \(95\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}}-\frac {9 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}\right )}{4}-2 a^{4} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )+2 a^{4} \left (\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\) | \(332\) |
1/24*(32*a^5*x^5-21*a^4*x^4-16*a^3*x^3+15*a^2*x^2-16*a*x+6)/x^4/(-c*(a^2*x ^2-1))^(1/2)*c+7/8*a^4*c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)
Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\left [\frac {21 \, a^{4} \sqrt {c} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac {21 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \]
[1/48*(21*a^4*sqrt(c)*x^4*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4, 1/24*(21*a^4*sqrt(-c)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a ^2*c*x^2 - c)) - (32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 1\right )}}{{\left (a x + 1\right )} x^{5}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.49 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {7 \, a^{4} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} - 96 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} + 128 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} {\left | a \right |} + 21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{4} - 32 \, a^{3} \sqrt {-c} c^{4} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \]
-7/4*a^4*c*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt( -c) + 1/12*(21*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^7*a^4*c - 45*(sqrt( -a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^4*c^2 - 96*(sqrt(-a^2*c)*x - sqrt(-a ^2*c*x^2 + c))^4*a^3*sqrt(-c)*c^2*abs(a) - 45*(sqrt(-a^2*c)*x - sqrt(-a^2* c*x^2 + c))^3*a^4*c^3 + 128*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^3* sqrt(-c)*c^3*abs(a) + 21*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^4*c^4 - 32*a^3*sqrt(-c)*c^4*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^4
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{x^5\,\left (a\,x+1\right )} \,d x \]