### 3.171 $$\int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx$$

Optimal. Leaf size=20 $\frac{1}{3} a^2 c^2 x^3-c^2 x$

[Out]

-(c^2*x) + (a^2*c^2*x^3)/3

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Rubi [A]  time = 0.0482971, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {6167, 6129, 41} $\frac{1}{3} a^2 c^2 x^3-c^2 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c^2*x) + (a^2*c^2*x^3)/3

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rubi steps

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

Mathematica [A]  time = 0.0077956, size = 17, normalized size = 0.85 $-c^2 \left (x-\frac{a^2 x^3}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c^2*(x - (a^2*x^3)/3))

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Maple [A]  time = 0.041, size = 17, normalized size = 0.9 \begin{align*}{c}^{2} \left ({\frac{{x}^{3}{a}^{2}}{3}}-x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06875, size = 24, normalized size = 1.2 \begin{align*} \frac{1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

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Fricas [A]  time = 1.41356, size = 34, normalized size = 1.7 \begin{align*} \frac{1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

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Sympy [A]  time = 0.080905, size = 15, normalized size = 0.75 \begin{align*} \frac{a^{2} c^{2} x^{3}}{3} - c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**2*x**3/3 - c**2*x

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Giac [A]  time = 1.13679, size = 24, normalized size = 1.2 \begin{align*} \frac{1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x