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

Optimal. Leaf size=52 \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{c^4 (a x+1)^8}{2 a}+\frac{4 c^4 (a x+1)^7}{7 a} \]

[Out]

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

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Rubi [A]  time = 0.0788417, antiderivative size = 52, 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} \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{c^4 (a x+1)^8}{2 a}+\frac{4 c^4 (a x+1)^7}{7 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0300276, size = 31, normalized size = 0.6 \[ \frac{c^4 (a x+1)^7 \left (14 a^2 x^2-35 a x+23\right )}{126 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(1 + a*x)^7*(23 - 35*a*x + 14*a^2*x^2))/(126*a)

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Maple [A]  time = 0.037, size = 69, normalized size = 1.3 \begin{align*}{c}^{4} \left ({\frac{{x}^{9}{a}^{8}}{9}}+{\frac{{a}^{7}{x}^{8}}{2}}+{\frac{4\,{x}^{7}{a}^{6}}{7}}-{\frac{2\,{x}^{6}{a}^{5}}{3}}-2\,{x}^{5}{a}^{4}-{x}^{4}{a}^{3}+{\frac{4\,{x}^{3}{a}^{2}}{3}}+2\,a{x}^{2}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x)

[Out]

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

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Maxima [A]  time = 1.11268, size = 124, normalized size = 2.38 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{2} \, a^{7} c^{4} x^{8} + \frac{4}{7} \, a^{6} c^{4} x^{7} - \frac{2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4}{3} \, a^{2} c^{4} x^{3} + 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)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47327, size = 190, normalized size = 3.65 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{2} \, a^{7} c^{4} x^{8} + \frac{4}{7} \, a^{6} c^{4} x^{7} - \frac{2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4}{3} \, a^{2} c^{4} x^{3} + 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)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.111963, size = 100, normalized size = 1.92 \begin{align*} \frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{2} + \frac{4 a^{6} c^{4} x^{7}}{7} - \frac{2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4 a^{2} c^{4} x^{3}}{3} + 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)**2*(a*x+1)**2*(-a**2*c*x**2+c)**4,x)

[Out]

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

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Giac [A]  time = 1.16992, size = 122, normalized size = 2.35 \begin{align*} \frac{{\left (14 \, c^{4} + \frac{189 \, c^{4}}{a x - 1} + \frac{1080 \, c^{4}}{{\left (a x - 1\right )}^{2}} + \frac{3360 \, c^{4}}{{\left (a x - 1\right )}^{3}} + \frac{6048 \, c^{4}}{{\left (a x - 1\right )}^{4}} + \frac{6048 \, c^{4}}{{\left (a x - 1\right )}^{5}} + \frac{2688 \, c^{4}}{{\left (a x - 1\right )}^{6}}\right )}{\left (a x - 1\right )}^{9}}{126 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

1/126*(14*c^4 + 189*c^4/(a*x - 1) + 1080*c^4/(a*x - 1)^2 + 3360*c^4/(a*x - 1)^3 + 6048*c^4/(a*x - 1)^4 + 6048*
c^4/(a*x - 1)^5 + 2688*c^4/(a*x - 1)^6)*(a*x - 1)^9/a