Integrand size = 27, antiderivative size = 123 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)^2}{3 a^2}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \arcsin (a x)}{a^2 \sqrt {1-a x} \sqrt {1+a x}} \]
x*(c-c/a^2/x^2)^(1/2)/a^2+1/3*x*(a*x+1)*(c-c/a^2/x^2)^(1/2)/a^2+1/3*x*(a*x +1)^2*(c-c/a^2/x^2)^(1/2)/a^2-x*arcsin(a*x)*(c-c/a^2/x^2)^(1/2)/a^2/(-a*x+ 1)^(1/2)/(a*x+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (5+3 a x+a^2 x^2\right )+3 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{3 a^2 \sqrt {-1+a^2 x^2}} \]
(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(5 + 3*a*x + a^2*x^2) + 3*Log [a*x + Sqrt[-1 + a^2*x^2]]))/(3*a^2*Sqrt[-1 + a^2*x^2])
Time = 0.59 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6717, 6709, 541, 25, 27, 533, 27, 455, 223}
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.
| \(\displaystyle \int x^2 \sqrt {c-\frac {c}{a^2 x^2}} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x (a x+1)^2}{\sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {\int -\frac {a^2 x (6 a x+5)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {\int \frac {a^2 x (6 a x+5)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} \int \frac {x (6 a x+5)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} \left (\frac {\int \frac {2 a (5 a x+3)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {3 x \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} \left (\frac {\int \frac {5 a x+3}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {3 x \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {5 \sqrt {1-a^2 x^2}}{a}}{a}-\frac {3 x \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {x \left (\frac {1}{3} \left (\frac {\frac {3 \arcsin (a x)}{a}-\frac {5 \sqrt {1-a^2 x^2}}{a}}{a}-\frac {3 x \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{3} x^2 \sqrt {1-a^2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
-((Sqrt[c - c/(a^2*x^2)]*x*(-1/3*(x^2*Sqrt[1 - a^2*x^2]) + ((-3*x*Sqrt[1 - a^2*x^2])/a + ((-5*Sqrt[1 - a^2*x^2])/a + (3*ArcSin[a*x])/a)/a)/3))/Sqrt[ 1 - a^2*x^2])
3.9.87.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 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 = 0.58 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(\frac {\left (a^{2} x^{2}+3 a x +5\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}{3 a^{2}}+\frac {\ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{a \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\) | \(124\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3}-3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x +3 c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-6 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right )-6 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c \right )}{3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c}\) | \(174\) |
1/3*(a^2*x^2+3*a*x+5)/a^2*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*x+1/a*ln(a^2*c*x/( a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)*x*(c*(a^2*x^2-1))^(1/2)*(c *(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)
Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.66 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\left [\frac {2 \, {\left (a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{6 \, a^{3}}, \frac {{\left (a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{3 \, a^{3}}\right ] \]
[1/6*(2*(a^3*x^3 + 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + 3* sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2) ) - c))/a^3, 1/3*((a^3*x^3 + 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2* x^2)) - 3*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) /(a^2*c*x^2 - c)))/a^3]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{a x - 1} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {1}{6} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (x {\left (\frac {x \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {3 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {5 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} - \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{3} {\left | a \right |}} + \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) - 10 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{4} {\left | a \right |}}\right )} {\left | a \right |} \]
1/6*(2*sqrt(a^2*c*x^2 - c)*(x*(x*sgn(x)/a^2 + 3*sgn(x)/a^3) + 5*sgn(x)/a^4 ) - 6*sqrt(c)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a^3*a bs(a)) + (3*a*sqrt(c)*log(abs(c)) - 10*sqrt(-c)*abs(a))*sgn(x)/(a^4*abs(a) ))*abs(a)
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]