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

Optimal. Leaf size=54 \[ \frac{c^2 (1-a x)^3}{3 a}+\frac{c^2 (1-a x)^2}{a}+\frac{8 c^2 \log (a x+1)}{a}-4 c^2 x \]

[Out]

-4*c^2*x + (c^2*(1 - a*x)^2)/a + (c^2*(1 - a*x)^3)/(3*a) + (8*c^2*Log[1 + a*x])/a

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Rubi [A]  time = 0.0363179, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac{c^2 (1-a x)^3}{3 a}+\frac{c^2 (1-a x)^2}{a}+\frac{8 c^2 \log (a x+1)}{a}-4 c^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4*c^2*x + (c^2*(1 - a*x)^2)/a + (c^2*(1 - a*x)^3)/(3*a) + (8*c^2*Log[1 + a*x])/a

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 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 e^{-2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c^2 \int \frac{(1-a x)^3}{1+a x} \, dx\\ &=c^2 \int \left (-4-2 (1-a x)-(1-a x)^2+\frac{8}{1+a x}\right ) \, dx\\ &=-4 c^2 x+\frac{c^2 (1-a x)^2}{a}+\frac{c^2 (1-a x)^3}{3 a}+\frac{8 c^2 \log (1+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0134649, size = 39, normalized size = 0.72 \[ -\frac{c^2 \left (a^3 x^3-6 a^2 x^2+21 a x-24 \log (a x+1)-4\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(-4 + 21*a*x - 6*a^2*x^2 + a^3*x^3 - 24*Log[1 + a*x]))/(3*a)

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Maple [A]  time = 0.029, size = 42, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}{c}^{2}{x}^{3}}{3}}+2\,{c}^{2}{x}^{2}a-7\,x{c}^{2}+8\,{\frac{{c}^{2}\ln \left ( ax+1 \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c^2*x^3+2*c^2*x^2*a-7*x*c^2+8*c^2*ln(a*x+1)/a

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Maxima [A]  time = 0.944377, size = 55, normalized size = 1.02 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} - 7 \, c^{2} x + \frac{8 \, c^{2} \log \left (a x + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c^2*x^3 + 2*a*c^2*x^2 - 7*c^2*x + 8*c^2*log(a*x + 1)/a

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Fricas [A]  time = 1.66764, size = 99, normalized size = 1.83 \begin{align*} -\frac{a^{3} c^{2} x^{3} - 6 \, a^{2} c^{2} x^{2} + 21 \, a c^{2} x - 24 \, c^{2} \log \left (a x + 1\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a^3*c^2*x^3 - 6*a^2*c^2*x^2 + 21*a*c^2*x - 24*c^2*log(a*x + 1))/a

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Sympy [A]  time = 0.3118, size = 41, normalized size = 0.76 \begin{align*} - \frac{a^{2} c^{2} x^{3}}{3} + 2 a c^{2} x^{2} - 7 c^{2} x + \frac{8 c^{2} \log{\left (a x + 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*c**2*x**3/3 + 2*a*c**2*x**2 - 7*c**2*x + 8*c**2*log(a*x + 1)/a

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Giac [A]  time = 1.17303, size = 92, normalized size = 1.7 \begin{align*} -\frac{{\left (c^{2} - \frac{9 \, c^{2}}{a x + 1} + \frac{36 \, c^{2}}{{\left (a x + 1\right )}^{2}}\right )}{\left (a x + 1\right )}^{3}}{3 \, a} - \frac{8 \, c^{2} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(c^2 - 9*c^2/(a*x + 1) + 36*c^2/(a*x + 1)^2)*(a*x + 1)^3/a - 8*c^2*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))
/a