### 3.169 $$\int e^{2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx$$

Optimal. Leaf size=37 $\frac{c^4 (1-a x)^4}{2 a}-\frac{c^4 (1-a x)^5}{5 a}$

[Out]

(c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*a)

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Rubi [A]  time = 0.0555531, antiderivative size = 37, 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} $\frac{c^4 (1-a x)^4}{2 a}-\frac{c^4 (1-a x)^5}{5 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0150451, size = 30, normalized size = 0.81 $\frac{1}{10} c^4 x \left (2 a^4 x^4-5 a^3 x^3+10 a x-10\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^4*x*(-10 + 10*a*x - 5*a^3*x^3 + 2*a^4*x^4))/10

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Maple [A]  time = 0.047, size = 30, normalized size = 0.8 \begin{align*}{c}^{4} \left ({\frac{{x}^{5}{a}^{4}}{5}}-{\frac{{x}^{4}{a}^{3}}{2}}+a{x}^{2}-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)^4,x)

[Out]

c^4*(1/5*x^5*a^4-1/2*x^4*a^3+a*x^2-x)

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Maxima [A]  time = 1.01339, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{1}{2} \, a^{3} c^{4} x^{4} + a c^{4} x^{2} - c^{4} 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)^4,x, algorithm="maxima")

[Out]

1/5*a^4*c^4*x^5 - 1/2*a^3*c^4*x^4 + a*c^4*x^2 - c^4*x

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Fricas [A]  time = 1.50833, size = 74, normalized size = 2. \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{1}{2} \, a^{3} c^{4} x^{4} + a c^{4} x^{2} - c^{4} 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)^4,x, algorithm="fricas")

[Out]

1/5*a^4*c^4*x^5 - 1/2*a^3*c^4*x^4 + a*c^4*x^2 - c^4*x

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Sympy [A]  time = 0.091233, size = 36, normalized size = 0.97 \begin{align*} \frac{a^{4} c^{4} x^{5}}{5} - \frac{a^{3} c^{4} x^{4}}{2} + a c^{4} x^{2} - c^{4} 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)**4,x)

[Out]

a**4*c**4*x**5/5 - a**3*c**4*x**4/2 + a*c**4*x**2 - c**4*x

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Giac [A]  time = 1.16564, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{1}{2} \, a^{3} c^{4} x^{4} + a c^{4} x^{2} - c^{4} 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)^4,x, algorithm="giac")

[Out]

1/5*a^4*c^4*x^5 - 1/2*a^3*c^4*x^4 + a*c^4*x^2 - c^4*x