3.1231 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=16 \[ -\frac{c (1-a x)^3}{3 a} \]

[Out]

-(c*(1 - a*x)^3)/(3*a)

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Rubi [A]  time = 0.0189674, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6140, 32} \[ -\frac{c (1-a x)^3}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(1 - a*x)^3)/(3*a)

Rule 6140

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0103685, size = 20, normalized size = 1.25 \[ c \left (\frac{a^2 x^3}{3}-a x^2+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(x - a*x^2 + (a^2*x^3)/3)

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Maple [A]  time = 0.026, size = 14, normalized size = 0.9 \begin{align*}{\frac{c \left ( ax-1 \right ) ^{3}}{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*(a*x-1)^3/a

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Maxima [A]  time = 0.950734, size = 27, normalized size = 1.69 \begin{align*} \frac{1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c*x^3 - a*c*x^2 + c*x

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Fricas [A]  time = 2.13448, size = 42, normalized size = 2.62 \begin{align*} \frac{1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c*x^3 - a*c*x^2 + c*x

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Sympy [A]  time = 0.150038, size = 19, normalized size = 1.19 \begin{align*} \frac{a^{2} c x^{3}}{3} - a c x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*x**3/3 - a*c*x**2 + c*x

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Giac [B]  time = 1.1363, size = 46, normalized size = 2.88 \begin{align*} \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{6 \, c}{a x + 1} + \frac{12 \, c}{{\left (a x + 1\right )}^{2}}\right )}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*x + 1)^3*(c - 6*c/(a*x + 1) + 12*c/(a*x + 1)^2)/a