3.1322 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \tanh ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]

[Out]

(2*E^(n*ArcTanh[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))

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Rubi [A]  time = 0.074394, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6136, 6137} \[ \frac{2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \tanh ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*E^(n*ArcTanh[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))

Rule 6136

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^2
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p
, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 6137

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

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}+\frac{2 \int \frac{e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c \left (4-n^2\right )}\\ &=\frac{2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0291504, size = 68, normalized size = 0.94 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}-1} \left (-2 a^2 x^2+2 a n x-n^2+2\right )}{a c^2 (n-2) n (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 55, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( 2\,{a}^{2}{x}^{2}-2\,nax+{n}^{2}-2 \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}an \left ({n}^{2}-4 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(n*arctanh(a*x))*(2*a^2*x^2-2*a*n*x+n^2-2)/(a^2*x^2-1)/c^2/a/n/(n^2-4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.34176, size = 163, normalized size = 2.26 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} - 2 \, a n x + n^{2} - 2\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n -{\left (a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))/(-a**2*c*x**2+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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