### 3.599 $$\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx$$

Optimal. Leaf size=55 $\frac{c^3 (1-a x)^7}{7 a}-\frac{2 c^3 (1-a x)^6}{3 a}+\frac{4 c^3 (1-a x)^5}{5 a}$

[Out]

(4*c^3*(1 - a*x)^5)/(5*a) - (2*c^3*(1 - a*x)^6)/(3*a) + (c^3*(1 - a*x)^7)/(7*a)

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Rubi [A]  time = 0.0754054, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {6167, 6140, 43} $\frac{c^3 (1-a x)^7}{7 a}-\frac{2 c^3 (1-a x)^6}{3 a}+\frac{4 c^3 (1-a x)^5}{5 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*c^3*(1 - a*x)^5)/(5*a) - (2*c^3*(1 - a*x)^6)/(3*a) + (c^3*(1 - a*x)^7)/(7*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 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 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)} \left (c-a^2 c x^2\right )^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\\ &=-\left (c^3 \int (1-a x)^4 (1+a x)^2 \, dx\right )\\ &=-\left (c^3 \int \left (4 (1-a x)^4-4 (1-a x)^5+(1-a x)^6\right ) \, dx\right )\\ &=\frac{4 c^3 (1-a x)^5}{5 a}-\frac{2 c^3 (1-a x)^6}{3 a}+\frac{c^3 (1-a x)^7}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0252143, size = 31, normalized size = 0.56 $-\frac{c^3 (a x-1)^5 \left (15 a^2 x^2+40 a x+29\right )}{105 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c^3*(-1 + a*x)^5*(29 + 40*a*x + 15*a^2*x^2))/(105*a)

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Maple [A]  time = 0.042, size = 54, normalized size = 1. \begin{align*}{c}^{3} \left ( -{\frac{{x}^{7}{a}^{6}}{7}}+{\frac{{x}^{6}{a}^{5}}{3}}+{\frac{{x}^{5}{a}^{4}}{5}}-{x}^{4}{a}^{3}+{\frac{{x}^{3}{a}^{2}}{3}}+a{x}^{2}-x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(-1/7*x^7*a^6+1/3*x^6*a^5+1/5*x^5*a^4-x^4*a^3+1/3*x^3*a^2+a*x^2-x)

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Maxima [A]  time = 1.00553, size = 95, normalized size = 1.73 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

-1/7*a^6*c^3*x^7 + 1/3*a^5*c^3*x^6 + 1/5*a^4*c^3*x^5 - a^3*c^3*x^4 + 1/3*a^2*c^3*x^3 + a*c^3*x^2 - c^3*x

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Fricas [A]  time = 1.48316, size = 143, normalized size = 2.6 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

-1/7*a^6*c^3*x^7 + 1/3*a^5*c^3*x^6 + 1/5*a^4*c^3*x^5 - a^3*c^3*x^4 + 1/3*a^2*c^3*x^3 + a*c^3*x^2 - c^3*x

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Sympy [A]  time = 0.095268, size = 70, normalized size = 1.27 \begin{align*} - \frac{a^{6} c^{3} x^{7}}{7} + \frac{a^{5} c^{3} x^{6}}{3} + \frac{a^{4} c^{3} x^{5}}{5} - a^{3} c^{3} x^{4} + \frac{a^{2} c^{3} x^{3}}{3} + a c^{3} x^{2} - c^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**3*(a*x-1)/(a*x+1),x)

[Out]

-a**6*c**3*x**7/7 + a**5*c**3*x**6/3 + a**4*c**3*x**5/5 - a**3*c**3*x**4 + a**2*c**3*x**3/3 + a*c**3*x**2 - c*
*3*x

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Giac [A]  time = 1.11891, size = 95, normalized size = 1.73 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^3*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

-1/7*a^6*c^3*x^7 + 1/3*a^5*c^3*x^6 + 1/5*a^4*c^3*x^5 - a^3*c^3*x^4 + 1/3*a^2*c^3*x^3 + a*c^3*x^2 - c^3*x