3.1321 \(\int \frac{e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{(2-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac{e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.149231, antiderivative size = 102, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6145, 6136, 6137} \[ \frac{n (n-2 a x) e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}-\frac{e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac{e^{n \tanh ^{-1}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6145

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

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

Mathematica [A]  time = 0.0340597, size = 56, normalized size = 0.81 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}-1} \left (a^2 x^2-a n x+1\right )}{a^2 c^2 \left (n^2-4\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 47, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ({a}^{2}{x}^{2}-nax+1 \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}{a}^{2} \left ({n}^{2}-4 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \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))*x/(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.2321, size = 151, normalized size = 2.19 \begin{align*} -\frac{{\left (a^{2} x^{2} - a n x + 1\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} -{\left (a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*x^2 - a*n*x + 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*n^2 - 4*a^2*c^2 - (a^4*c^2*n^2 - 4*a^4*c^2)*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))*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{x \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))*x/(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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