### 3.208 $$\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx$$

Optimal. Leaf size=73 $-\frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{2 c^3 (1-a x)^2}{a}-\frac{16 c^3 \log (a x+1)}{a}+8 c^3 x$

[Out]

8*c^3*x - (2*c^3*(1 - a*x)^2)/a - (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a) - (16*c^3*Log[1 + a*x])/
a

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Rubi [A]  time = 0.0645011, antiderivative size = 73, 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, 43} $-\frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{2 c^3 (1-a x)^2}{a}-\frac{16 c^3 \log (a x+1)}{a}+8 c^3 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

8*c^3*x - (2*c^3*(1 - a*x)^2)/a - (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a) - (16*c^3*Log[1 + a*x])/
a

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 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 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=-\left (c^3 \int \frac{(1-a x)^4}{1+a x} \, dx\right )\\ &=-\left (c^3 \int \left (-8-4 (1-a x)-2 (1-a x)^2-(1-a x)^3+\frac{16}{1+a x}\right ) \, dx\right )\\ &=8 c^3 x-\frac{2 c^3 (1-a x)^2}{a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a}-\frac{16 c^3 \log (1+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0176466, size = 48, normalized size = 0.66 $-\frac{c^3 \left (3 a^4 x^4-20 a^3 x^3+66 a^2 x^2-180 a x+192 \log (a x+1)+35\right )}{12 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c^3*(35 - 180*a*x + 66*a^2*x^2 - 20*a^3*x^3 + 3*a^4*x^4 + 192*Log[1 + a*x]))/(12*a)

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Maple [A]  time = 0.043, size = 53, normalized size = 0.7 \begin{align*} -{\frac{{c}^{3}{x}^{4}{a}^{3}}{4}}+{\frac{5\,{c}^{3}{x}^{3}{a}^{2}}{3}}-{\frac{11\,{c}^{3}{x}^{2}a}{2}}+15\,{c}^{3}x-16\,{\frac{{c}^{3}\ln \left ( ax+1 \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^3*x^4*a^3+5/3*c^3*x^3*a^2-11/2*c^3*x^2*a+15*c^3*x-16*c^3*ln(a*x+1)/a

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Maxima [A]  time = 1.03387, size = 70, normalized size = 0.96 \begin{align*} -\frac{1}{4} \, a^{3} c^{3} x^{4} + \frac{5}{3} \, a^{2} c^{3} x^{3} - \frac{11}{2} \, a c^{3} x^{2} + 15 \, c^{3} x - \frac{16 \, c^{3} \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)^3*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

-1/4*a^3*c^3*x^4 + 5/3*a^2*c^3*x^3 - 11/2*a*c^3*x^2 + 15*c^3*x - 16*c^3*log(a*x + 1)/a

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Fricas [A]  time = 1.45557, size = 130, normalized size = 1.78 \begin{align*} -\frac{3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log \left (a x + 1\right )}{12 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*c^3*x^4 - 20*a^3*c^3*x^3 + 66*a^2*c^3*x^2 - 180*a*c^3*x + 192*c^3*log(a*x + 1))/a

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Sympy [A]  time = 0.601988, size = 56, normalized size = 0.77 \begin{align*} - \frac{a^{3} c^{3} x^{4}}{4} + \frac{5 a^{2} c^{3} x^{3}}{3} - \frac{11 a c^{3} x^{2}}{2} + 15 c^{3} x - \frac{16 c^{3} \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)**3*(a*x-1)/(a*x+1),x)

[Out]

-a**3*c**3*x**4/4 + 5*a**2*c**3*x**3/3 - 11*a*c**3*x**2/2 + 15*c**3*x - 16*c**3*log(a*x + 1)/a

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Giac [A]  time = 1.12859, size = 86, normalized size = 1.18 \begin{align*} -\frac{16 \, c^{3} \log \left ({\left | a x + 1 \right |}\right )}{a} - \frac{3 \, a^{7} c^{3} x^{4} - 20 \, a^{6} c^{3} x^{3} + 66 \, a^{5} c^{3} x^{2} - 180 \, a^{4} c^{3} x}{12 \, a^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*c^3*log(abs(a*x + 1))/a - 1/12*(3*a^7*c^3*x^4 - 20*a^6*c^3*x^3 + 66*a^5*c^3*x^2 - 180*a^4*c^3*x)/a^4