3.1050 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=53 \[ \frac{x^2}{2 a^2 c}+\frac{2 x}{a^3 c}+\frac{1}{a^4 c (1-a x)}+\frac{3 \log (1-a x)}{a^4 c} \]

[Out]

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

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Rubi [A]  time = 0.10409, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 43} \[ \frac{x^2}{2 a^2 c}+\frac{2 x}{a^3 c}+\frac{1}{a^4 c (1-a x)}+\frac{3 \log (1-a x)}{a^4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0299233, size = 41, normalized size = 0.77 \[ \frac{a^2 x^2+4 a x+\frac{2}{1-a x}+6 \log (1-a x)}{2 a^4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*x + a^2*x^2 + 2/(1 - a*x) + 6*Log[1 - a*x])/(2*a^4*c)

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Maple [A]  time = 0.034, size = 51, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}c}}+2\,{\frac{x}{{a}^{3}c}}-{\frac{1}{c{a}^{4} \left ( ax-1 \right ) }}+3\,{\frac{\ln \left ( ax-1 \right ) }{c{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965163, size = 66, normalized size = 1.25 \begin{align*} -\frac{1}{a^{5} c x - a^{4} c} + \frac{a x^{2} + 4 \, x}{2 \, a^{3} c} + \frac{3 \, \log \left (a x - 1\right )}{a^{4} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90143, size = 113, normalized size = 2.13 \begin{align*} \frac{a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 6 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{2 \,{\left (a^{5} c x - a^{4} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.35384, size = 44, normalized size = 0.83 \begin{align*} - \frac{1}{a^{5} c x - a^{4} c} + \frac{x^{2}}{2 a^{2} c} + \frac{2 x}{a^{3} c} + \frac{3 \log{\left (a x - 1 \right )}}{a^{4} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16419, size = 72, normalized size = 1.36 \begin{align*} \frac{3 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4} c} + \frac{a^{2} c x^{2} + 4 \, a c x}{2 \, a^{4} c^{2}} - \frac{1}{{\left (a x - 1\right )} a^{4} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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