\(\int \frac {e^{\coth ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{3/2}} \, dx\) [693]

Optimal result

Integrand size = 25, antiderivative size = 130 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6327, 6328, 90} \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

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 {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {x^2}{(-1+a x)^2 (1+a x)} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (\frac {1}{2 a^2 (-1+a x)^2}+\frac {3}{4 a^2 (-1+a x)}+\frac {1}{4 a^2 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (\frac {2}{1-a x}+3 \log (1-a x)+\log (1+a x)\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a \ln \left (a x +1\right ) x +3 a \ln \left (a x -1\right ) x -\ln \left (a x +1\right )-3 \ln \left (a x -1\right )-2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} a^{3}}\)	\(86\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {-a^{2} c} {\left ({\left (a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )}}{4 \, {\left (a^{5} c^{2} x - a^{4} c^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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