Integrand size = 27, antiderivative size = 152 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \] Output:
4*(c-c/a^2/x^2)^(1/2)*x/a^2/(1-1/a^2/x^2)^(1/2)+3/2*(c-c/a^2/x^2)^(1/2)*x^ 2/a/(1-1/a^2/x^2)^(1/2)+1/3*(c-c/a^2/x^2)^(1/2)*x^3/(1-1/a^2/x^2)^(1/2)+4* (c-c/a^2/x^2)^(1/2)*ln(-a*x+1)/a^3/(1-1/a^2/x^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.41 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (a x \left (24+9 a x+2 a^2 x^2\right )+24 \log (1-a x)\right )}{6 a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \] Input:
Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*(a*x*(24 + 9*a*x + 2*a^2*x^2) + 24*Log[1 - a*x]))/( 6*a^3*Sqrt[1 - 1/(a^2*x^2)])
Time = 0.89 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.45, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6751, 6747, 25, 86, 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.
\(\displaystyle \int x^2 \sqrt {c-\frac {c}{a^2 x^2}} e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^2dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int -\frac {x (a x+1)^2}{1-a x}dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x (a x+1)^2}{1-a x}dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (-a x^2-3 x-\frac {4}{a}-\frac {4}{a (a x-1)}\right )dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {4 \log (1-a x)}{a^2}-\frac {a x^3}{3}-\frac {4 x}{a}-\frac {3 x^2}{2}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2,x]
Output:
-((Sqrt[c - c/(a^2*x^2)]*((-4*x)/a - (3*x^2)/2 - (a*x^3)/3 - (4*Log[1 - a* x])/a^2))/(a*Sqrt[1 - 1/(a^2*x^2)]))
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart [p]) Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54
method | result | size |
default | \(\frac {\left (2 a^{3} x^{3}+9 a^{2} x^{2}+24 a x +24 \ln \left (a x -1\right )\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x -1\right )}{6 a^{2} \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(82\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)*x^2,x,method=_RETURNVERB OSE)
Output:
1/6*(2*a^3*x^3+9*a^2*x^2+24*a*x+24*ln(a*x-1))*x*(c*(a^2*x^2-1)/a^2/x^2)^(1 /2)*(a*x-1)/a^2/(a*x+1)^2/((a*x-1)/(a*x+1))^(3/2)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {{\left (2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x + 24 \, \log \left (a x - 1\right )\right )} \sqrt {a^{2} c}}{6 \, a^{4}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)*x^2,x, algorithm=" fricas")
Output:
1/6*(2*a^3*x^3 + 9*a^2*x^2 + 24*a*x + 24*log(a*x - 1))*sqrt(a^2*c)/a^4
Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a**2/x**2)**(1/2)*x**2,x)
Output:
Timed out
\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)*x^2,x, algorithm=" maxima")
Output:
integrate(sqrt(c - c/(a^2*x^2))*x^2/((a*x - 1)/(a*x + 1))^(3/2), x)
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {1}{6} \, \sqrt {c} {\left (\frac {2 \, a^{2} x^{3} \mathrm {sgn}\left (x\right ) + 9 \, a x^{2} \mathrm {sgn}\left (x\right ) + 24 \, x \mathrm {sgn}\left (x\right )}{a^{3} \mathrm {sgn}\left (a x + 1\right )} + \frac {48 \, \log \left ({\left | a x - 1 \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{4} \mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, a^{8} x^{3} \mathrm {sgn}\left (x\right ) + 9 \, a^{7} x^{2} \mathrm {sgn}\left (x\right ) + 24 \, a^{6} x \mathrm {sgn}\left (x\right )}{a^{9} \mathrm {sgn}\left (a x + 1\right )}\right )} {\left | a \right |} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)*x^2,x, algorithm=" giac")
Output:
1/6*sqrt(c)*((2*a^2*x^3*sgn(x) + 9*a*x^2*sgn(x) + 24*x*sgn(x))/(a^3*sgn(a* x + 1)) + 48*log(abs(a*x - 1))*sgn(x)/(a^4*sgn(a*x + 1)) + (2*a^8*x^3*sgn( x) + 9*a^7*x^2*sgn(x) + 24*a^6*x*sgn(x))/(a^9*sgn(a*x + 1)))*abs(a)
Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int((x^2*(c - c/(a^2*x^2))^(1/2))/((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
int((x^2*(c - c/(a^2*x^2))^(1/2))/((a*x - 1)/(a*x + 1))^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c}\, \left (24 \,\mathrm {log}\left (a x -1\right )+2 a^{3} x^{3}+9 a^{2} x^{2}+24 a x \right )}{6 a^{3}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)*x^2,x)
Output:
(sqrt(c)*(24*log(a*x - 1) + 2*a**3*x**3 + 9*a**2*x**2 + 24*a*x))/(6*a**3)