### 3.207 $$\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx$$

Optimal. Leaf size=91 $-\frac{c^4 (1-a x)^5}{5 a}-\frac{c^4 (1-a x)^4}{2 a}-\frac{4 c^4 (1-a x)^3}{3 a}-\frac{4 c^4 (1-a x)^2}{a}-\frac{32 c^4 \log (a x+1)}{a}+16 c^4 x$

[Out]

16*c^4*x - (4*c^4*(1 - a*x)^2)/a - (4*c^4*(1 - a*x)^3)/(3*a) - (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*
a) - (32*c^4*Log[1 + a*x])/a

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Rubi [A]  time = 0.0706506, antiderivative size = 91, 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} $-\frac{c^4 (1-a x)^5}{5 a}-\frac{c^4 (1-a x)^4}{2 a}-\frac{4 c^4 (1-a x)^3}{3 a}-\frac{4 c^4 (1-a x)^2}{a}-\frac{32 c^4 \log (a x+1)}{a}+16 c^4 x$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - a*c*x)^4/E^(2*ArcCoth[a*x]),x]

[Out]

16*c^4*x - (4*c^4*(1 - a*x)^2)/a - (4*c^4*(1 - a*x)^3)/(3*a) - (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*
a) - (32*c^4*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)^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=-\left (c^4 \int \frac{(1-a x)^5}{1+a x} \, dx\right )\\ &=-\left (c^4 \int \left (-16-8 (1-a x)-4 (1-a x)^2-2 (1-a x)^3-(1-a x)^4+\frac{32}{1+a x}\right ) \, dx\right )\\ &=16 c^4 x-\frac{4 c^4 (1-a x)^2}{a}-\frac{4 c^4 (1-a x)^3}{3 a}-\frac{c^4 (1-a x)^4}{2 a}-\frac{c^4 (1-a x)^5}{5 a}-\frac{32 c^4 \log (1+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0205326, size = 56, normalized size = 0.62 $\frac{c^4 \left (6 a^5 x^5-45 a^4 x^4+160 a^3 x^3-390 a^2 x^2+930 a x-960 \log (a x+1)-181\right )}{30 a}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c - a*c*x)^4/E^(2*ArcCoth[a*x]),x]

[Out]

(c^4*(-181 + 930*a*x - 390*a^2*x^2 + 160*a^3*x^3 - 45*a^4*x^4 + 6*a^5*x^5 - 960*Log[1 + a*x]))/(30*a)

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Maple [A]  time = 0.04, size = 64, normalized size = 0.7 \begin{align*}{\frac{{a}^{4}{c}^{4}{x}^{5}}{5}}-{\frac{3\,{c}^{4}{x}^{4}{a}^{3}}{2}}+{\frac{16\,{a}^{2}{c}^{4}{x}^{3}}{3}}-13\,{c}^{4}{x}^{2}a+31\,{c}^{4}x-32\,{\frac{{c}^{4}\ln \left ( ax+1 \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^4/(a*x+1)*(a*x-1),x)

[Out]

1/5*a^4*c^4*x^5-3/2*c^4*x^4*a^3+16/3*a^2*c^4*x^3-13*c^4*x^2*a+31*c^4*x-32*c^4*ln(a*x+1)/a

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Maxima [A]  time = 1.00712, size = 85, normalized size = 0.93 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{16}{3} \, a^{2} c^{4} x^{3} - 13 \, a c^{4} x^{2} + 31 \, c^{4} x - \frac{32 \, c^{4} \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)^4*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

1/5*a^4*c^4*x^5 - 3/2*a^3*c^4*x^4 + 16/3*a^2*c^4*x^3 - 13*a*c^4*x^2 + 31*c^4*x - 32*c^4*log(a*x + 1)/a

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Fricas [A]  time = 1.51959, size = 154, normalized size = 1.69 \begin{align*} \frac{6 \, a^{5} c^{4} x^{5} - 45 \, a^{4} c^{4} x^{4} + 160 \, a^{3} c^{4} x^{3} - 390 \, a^{2} c^{4} x^{2} + 930 \, a c^{4} x - 960 \, c^{4} \log \left (a x + 1\right )}{30 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

1/30*(6*a^5*c^4*x^5 - 45*a^4*c^4*x^4 + 160*a^3*c^4*x^3 - 390*a^2*c^4*x^2 + 930*a*c^4*x - 960*c^4*log(a*x + 1))
/a

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Sympy [A]  time = 0.670782, size = 68, normalized size = 0.75 \begin{align*} \frac{a^{4} c^{4} x^{5}}{5} - \frac{3 a^{3} c^{4} x^{4}}{2} + \frac{16 a^{2} c^{4} x^{3}}{3} - 13 a c^{4} x^{2} + 31 c^{4} x - \frac{32 c^{4} \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)**4*(a*x-1)/(a*x+1),x)

[Out]

a**4*c**4*x**5/5 - 3*a**3*c**4*x**4/2 + 16*a**2*c**4*x**3/3 - 13*a*c**4*x**2 + 31*c**4*x - 32*c**4*log(a*x + 1
)/a

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Giac [A]  time = 1.19164, size = 101, normalized size = 1.11 \begin{align*} -\frac{32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{6 \, a^{9} c^{4} x^{5} - 45 \, a^{8} c^{4} x^{4} + 160 \, a^{7} c^{4} x^{3} - 390 \, a^{6} c^{4} x^{2} + 930 \, a^{5} c^{4} x}{30 \, a^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

-32*c^4*log(abs(a*x + 1))/a + 1/30*(6*a^9*c^4*x^5 - 45*a^8*c^4*x^4 + 160*a^7*c^4*x^3 - 390*a^6*c^4*x^2 + 930*a
^5*c^4*x)/a^5