\(\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx\) [721]

Optimal result

Integrand size = 27, antiderivative size = 227 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[Out]

4*(-a^2*c*x^2+c)^(1/2)/a^4/(1-1/a^2/x^2)^(1/2)-2*x*(-a^2*c*x^2+c)^(1/2)/a^3/(1-1/a^2/x^2)^(1/2)+4/3*x^2*(-a^2*
c*x^2+c)^(1/2)/a^2/(1-1/a^2/x^2)^(1/2)-3/4*x^3*(-a^2*c*x^2+c)^(1/2)/a/(1-1/a^2/x^2)^(1/2)+1/5*x^4*(-a^2*c*x^2+
c)^(1/2)/(1-1/a^2/x^2)^(1/2)-4*ln(a*x+1)*(-a^2*c*x^2+c)^(1/2)/a^5/x/(1-1/a^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 90} \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a^5 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

Int[(x^3*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]

[Out]

(4*Sqrt[c - a^2*c*x^2])/(a^4*Sqrt[1 - 1/(a^2*x^2)]) - (2*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - 1/(a^2*x^2)]) +
(4*x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]) - (3*x^3*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - 1/(a^2*x
^2)]) + (x^4*Sqrt[c - a^2*c*x^2])/(5*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^5*Sqrt[1
- 1/(a^2*x^2)]*x)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rule 6327

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c + d*x^2)^p/(x^(2*p)*(
1 - 1/(a^2*x^2))^p), Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x]
&& EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[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] &&  !
IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^4 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {x^3 (-1+a x)^2}{1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (\frac {4}{a^3}-\frac {4 x}{a^2}+\frac {4 x^2}{a}-3 x^3+a x^4-\frac {4}{a^3 (1+a x)}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.38 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\frac {4 x}{a^4}-\frac {2 x^2}{a^3}+\frac {4 x^3}{3 a^2}-\frac {3 x^4}{4 a}+\frac {x^5}{5}-\frac {4 \log (1+a x)}{a^5}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]

[In]

Integrate[(x^3*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]

[Out]

(Sqrt[c - a^2*c*x^2]*((4*x)/a^4 - (2*x^2)/a^3 + (4*x^3)/(3*a^2) - (3*x^4)/(4*a) + x^5/5 - (4*Log[1 + a*x])/a^5
))/(Sqrt[1 - 1/(a^2*x^2)]*x)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.41

method	result	size
default	\(-\frac {\left (-12 a^{5} x^{5}+45 a^{4} x^{4}-80 a^{3} x^{3}+120 a^{2} x^{2}-240 a x +240 \ln \left (a x +1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{60 \left (a x -1\right )^{2} a^{4}}\)	\(92\)

[In]

int(x^3*(-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/60*(-12*a^5*x^5+45*a^4*x^4-80*a^3*x^3+120*a^2*x^2-240*a*x+240*ln(a*x+1))*(-c*(a^2*x^2-1))^(1/2)*(a*x+1)*((a
*x-1)/(a*x+1))^(3/2)/(a*x-1)^2/a^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.26 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x - 240 \, \log \left (a x + 1\right )\right )} \sqrt {-a^{2} c}}{60 \, a^{5}} \]

[In]

integrate(x^3*(-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

1/60*(12*a^5*x^5 - 45*a^4*x^4 + 80*a^3*x^3 - 120*a^2*x^2 + 240*a*x - 240*log(a*x + 1))*sqrt(-a^2*c)/a^5

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\text {Timed out} \]

[In]

integrate(x**3*(-a**2*c*x**2+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x^3*(-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*x^3*((a*x - 1)/(a*x + 1))^(3/2), x)

Giac [F]

\[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x^3*(-a^2*c*x^2+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*x^3*((a*x - 1)/(a*x + 1))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int x^3\,\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]

[In]

int(x^3*(c - a^2*c*x^2)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

int(x^3*(c - a^2*c*x^2)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2), x)