### 3.565 $$\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx$$

Optimal. Leaf size=69 $\frac{c^4 (a x+1)^9}{9 a}-\frac{3 c^4 (a x+1)^8}{4 a}+\frac{12 c^4 (a x+1)^7}{7 a}-\frac{4 c^4 (a x+1)^6}{3 a}$

[Out]

(-4*c^4*(1 + a*x)^6)/(3*a) + (12*c^4*(1 + a*x)^7)/(7*a) - (3*c^4*(1 + a*x)^8)/(4*a) + (c^4*(1 + a*x)^9)/(9*a)

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Rubi [A]  time = 0.0850346, antiderivative size = 69, 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} $\frac{c^4 (a x+1)^9}{9 a}-\frac{3 c^4 (a x+1)^8}{4 a}+\frac{12 c^4 (a x+1)^7}{7 a}-\frac{4 c^4 (a x+1)^6}{3 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-4*c^4*(1 + a*x)^6)/(3*a) + (12*c^4*(1 + a*x)^7)/(7*a) - (3*c^4*(1 + a*x)^8)/(4*a) + (c^4*(1 + a*x)^9)/(9*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 )^4 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\\ &=-\left (c^4 \int (1-a x)^3 (1+a x)^5 \, dx\right )\\ &=-\left (c^4 \int \left (8 (1+a x)^5-12 (1+a x)^6+6 (1+a x)^7-(1+a x)^8\right ) \, dx\right )\\ &=-\frac{4 c^4 (1+a x)^6}{3 a}+\frac{12 c^4 (1+a x)^7}{7 a}-\frac{3 c^4 (1+a x)^8}{4 a}+\frac{c^4 (1+a x)^9}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.0315883, size = 39, normalized size = 0.57 $\frac{c^4 (a x+1)^6 \left (28 a^3 x^3-105 a^2 x^2+138 a x-65\right )}{252 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^4*(1 + a*x)^6*(-65 + 138*a*x - 105*a^2*x^2 + 28*a^3*x^3))/(252*a)

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Maple [A]  time = 0.038, size = 63, normalized size = 0.9 \begin{align*}{c}^{4} \left ({\frac{{x}^{9}{a}^{8}}{9}}+{\frac{{a}^{7}{x}^{8}}{4}}-{\frac{2\,{x}^{7}{a}^{6}}{7}}-{x}^{6}{a}^{5}+{\frac{3\,{x}^{4}{a}^{3}}{2}}+{\frac{2\,{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*x+1)/(a*x-1)*(-a^2*c*x^2+c)^4,x)

[Out]

c^4*(1/9*x^9*a^8+1/4*a^7*x^8-2/7*x^7*a^6-x^6*a^5+3/2*x^4*a^3+2/3*x^3*a^2-a*x^2-x)

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Maxima [A]  time = 1.21455, size = 111, normalized size = 1.61 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} - 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^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 - a*c^4*
x^2 - c^4*x

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Fricas [A]  time = 1.55152, size = 166, normalized size = 2.41 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} - 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^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 - a*c^4*
x^2 - c^4*x

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Sympy [A]  time = 0.100179, size = 87, normalized size = 1.26 \begin{align*} \frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{4} - \frac{2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac{3 a^{3} c^{4} x^{4}}{2} + \frac{2 a^{2} c^{4} x^{3}}{3} - 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**2*c*x**2+c)**4,x)

[Out]

a**8*c**4*x**9/9 + a**7*c**4*x**8/4 - 2*a**6*c**4*x**7/7 - a**5*c**4*x**6 + 3*a**3*c**4*x**4/2 + 2*a**2*c**4*x
**3/3 - a*c**4*x**2 - c**4*x

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Giac [A]  time = 1.1178, size = 111, normalized size = 1.61 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} - 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^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 - 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 + 2/3*a^2*c^4*x^3 - a*c^4*
x^2 - c^4*x