3.1343 \(\int \frac{e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{(1-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0932925, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {6144} \[ \frac{(1-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6144

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

Rubi steps

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

Mathematica [A]  time = 0.0576016, size = 81, normalized size = 1.76 \[ \frac{\sqrt{1-a^2 x^2} (1-a x)^{-\frac{n}{2}-\frac{1}{2}} (a x+1)^{\frac{n-1}{2}} (a n x-1)}{a^2 c (n-1) (n+1) \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 49, 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}^{2} \left ({n}^{2}-1 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \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)^(3/2),x)

[Out]

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

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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 )}^{\frac{3}{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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.25363, size = 159, normalized size = 3.46 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a n x - 1\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} -{\left (a^{4} c^{2} n^{2} - 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{n \operatorname{atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral(x*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)

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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 )}^{\frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

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