3.1367 \(\int e^{n \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^{-1-\frac{n^2}{2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{(1-a n x) \left (c-a^2 c x^2\right )^{-\frac{n^2}{2}} e^{n \tanh ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

[Out]

(E^(n*ArcTanh[a*x])*(1 - a*n*x))/(a^3*c*n*(1 - n^2)*(c - a^2*c*x^2)^(n^2/2))

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Rubi [A]  time = 0.109377, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {6146} \[ \frac{(1-a n x) \left (c-a^2 c x^2\right )^{-\frac{n^2}{2}} e^{n \tanh ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^(-1 - n^2/2),x]

[Out]

(E^(n*ArcTanh[a*x])*(1 - a*n*x))/(a^3*c*n*(1 - n^2)*(c - a^2*c*x^2)^(n^2/2))

Rule 6146

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

Rubi steps

\begin{align*} \int e^{n \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac{n^2}{2}} \, dx &=\frac{e^{n \tanh ^{-1}(a x)} (1-a n x) \left (c-a^2 c x^2\right )^{-\frac{n^2}{2}}}{a^3 c n \left (1-n^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0735886, size = 92, normalized size = 1.74 \[ \frac{(1-a x)^{-\frac{1}{2} n (n+1)} (a x+1)^{-\frac{1}{2} (n-1) n} (a n x-1) \left (1-a^2 x^2\right )^{\frac{n^2}{2}} \left (c-a^2 c x^2\right )^{-\frac{n^2}{2}}}{a^3 c (n-1) n (n+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^(-1 - n^2/2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 58, normalized size = 1.1 \begin{align*} -{\frac{ \left ( nax-1 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{a}^{3}n \left ({n}^{2}-1 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-1-{\frac{{n}^{2}}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x)

[Out]

-(a*x-1)*(a*x+1)*(a*n*x-1)*exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(-1-1/2*n^2)/a^3/n/(n^2-1)

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Maxima [A]  time = 1.02798, size = 101, normalized size = 1.91 \begin{align*} \frac{{\left (a n x - 1\right )} e^{\left (-\frac{1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac{1}{2} \, n^{2} \log \left (a x - 1\right ) + \frac{1}{2} \, n \log \left (a x + 1\right ) - \frac{1}{2} \, n \log \left (a x - 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3} \left (-c\right )^{\frac{1}{2} \, n^{2}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*n*x - 1)*e^(-1/2*n^2*log(a*x + 1) - 1/2*n^2*log(a*x - 1) + 1/2*n*log(a*x + 1) - 1/2*n*log(a*x - 1))/((n^3 -
n)*a^3*(-c)^(1/2*n^2)*c)

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Fricas [A]  time = 2.18517, size = 157, normalized size = 2.96 \begin{align*} -\frac{{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{-\frac{1}{2} \, n^{2} - 1} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{3} n^{3} - a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))*x**2*(-a**2*c*x**2+c)**(-1-1/2*n**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{-\frac{1}{2} \, n^{2} - 1} x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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