3.920 \(\int \frac{e^{\tanh ^{-1}(a x)} x^4}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]

[Out]

-(x/a^4) - x^2/(2*a^3) - x^3/(3*a^2) - x^4/(4*a) - Log[1 - a*x]/a^5

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Rubi [A]  time = 0.0999109, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 43} \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^4)/Sqrt[1 - a^2*x^2],x]

[Out]

-(x/a^4) - x^2/(2*a^3) - x^3/(3*a^2) - x^4/(4*a) - Log[1 - a*x]/a^5

Rule 6150

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{x^4}{1-a x} \, dx\\ &=\int \left (-\frac{1}{a^4}-\frac{x}{a^3}-\frac{x^2}{a^2}-\frac{x^3}{a}-\frac{1}{a^4 (-1+a x)}\right ) \, dx\\ &=-\frac{x}{a^4}-\frac{x^2}{2 a^3}-\frac{x^3}{3 a^2}-\frac{x^4}{4 a}-\frac{\log (1-a x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.024949, size = 49, normalized size = 1. \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^4)/Sqrt[1 - a^2*x^2],x]

[Out]

-(x/a^4) - x^2/(2*a^3) - x^3/(3*a^2) - x^4/(4*a) - Log[1 - a*x]/a^5

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Maple [A]  time = 0.029, size = 43, normalized size = 0.9 \begin{align*} -{\frac{{x}^{4}}{4\,a}}-{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{x}^{2}}{2\,{a}^{3}}}-{\frac{x}{{a}^{4}}}-{\frac{\ln \left ( ax-1 \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)*x^4,x)

[Out]

-1/4*x^4/a-1/3*x^3/a^2-1/2*x^2/a^3-x/a^4-1/a^5*ln(a*x-1)

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Maxima [A]  time = 0.9511, size = 58, normalized size = 1.18 \begin{align*} -\frac{3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac{\log \left (a x - 1\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)*x^4,x, algorithm="maxima")

[Out]

-1/12*(3*a^3*x^4 + 4*a^2*x^3 + 6*a*x^2 + 12*x)/a^4 - log(a*x - 1)/a^5

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Fricas [A]  time = 1.45332, size = 100, normalized size = 2.04 \begin{align*} -\frac{3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x + 12 \, \log \left (a x - 1\right )}{12 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)*x^4,x, algorithm="fricas")

[Out]

-1/12*(3*a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 12*a*x + 12*log(a*x - 1))/a^5

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Sympy [A]  time = 0.263434, size = 39, normalized size = 0.8 \begin{align*} - \frac{x^{4}}{4 a} - \frac{x^{3}}{3 a^{2}} - \frac{x^{2}}{2 a^{3}} - \frac{x}{a^{4}} - \frac{\log{\left (a x - 1 \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)*x**4,x)

[Out]

-x**4/(4*a) - x**3/(3*a**2) - x**2/(2*a**3) - x/a**4 - log(a*x - 1)/a**5

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Giac [A]  time = 1.21649, size = 59, normalized size = 1.2 \begin{align*} -\frac{3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)*x^4,x, algorithm="giac")

[Out]

-1/12*(3*a^3*x^4 + 4*a^2*x^3 + 6*a*x^2 + 12*x)/a^4 - log(abs(a*x - 1))/a^5