### 3.564 $$\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^5 \, dx$$

Optimal. Leaf size=84 $-\frac{c^5 (a x+1)^{11}}{11 a}+\frac{4 c^5 (a x+1)^{10}}{5 a}-\frac{8 c^5 (a x+1)^9}{3 a}+\frac{4 c^5 (a x+1)^8}{a}-\frac{16 c^5 (a x+1)^7}{7 a}$

[Out]

(-16*c^5*(1 + a*x)^7)/(7*a) + (4*c^5*(1 + a*x)^8)/a - (8*c^5*(1 + a*x)^9)/(3*a) + (4*c^5*(1 + a*x)^10)/(5*a) -
(c^5*(1 + a*x)^11)/(11*a)

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Rubi [A]  time = 0.0988688, antiderivative size = 84, 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} $-\frac{c^5 (a x+1)^{11}}{11 a}+\frac{4 c^5 (a x+1)^{10}}{5 a}-\frac{8 c^5 (a x+1)^9}{3 a}+\frac{4 c^5 (a x+1)^8}{a}-\frac{16 c^5 (a x+1)^7}{7 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

(-16*c^5*(1 + a*x)^7)/(7*a) + (4*c^5*(1 + a*x)^8)/a - (8*c^5*(1 + a*x)^9)/(3*a) + (4*c^5*(1 + a*x)^10)/(5*a) -
(c^5*(1 + a*x)^11)/(11*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 )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\\ &=-\left (c^5 \int (1-a x)^4 (1+a x)^6 \, dx\right )\\ &=-\left (c^5 \int \left (16 (1+a x)^6-32 (1+a x)^7+24 (1+a x)^8-8 (1+a x)^9+(1+a x)^{10}\right ) \, dx\right )\\ &=-\frac{16 c^5 (1+a x)^7}{7 a}+\frac{4 c^5 (1+a x)^8}{a}-\frac{8 c^5 (1+a x)^9}{3 a}+\frac{4 c^5 (1+a x)^{10}}{5 a}-\frac{c^5 (1+a x)^{11}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0411871, size = 47, normalized size = 0.56 $-\frac{c^5 (a x+1)^7 \left (105 a^4 x^4-504 a^3 x^3+938 a^2 x^2-812 a x+281\right )}{1155 a}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

-(c^5*(1 + a*x)^7*(281 - 812*a*x + 938*a^2*x^2 - 504*a^3*x^3 + 105*a^4*x^4))/(1155*a)

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Maple [A]  time = 0.037, size = 85, normalized size = 1. \begin{align*}{c}^{5} \left ( -{\frac{{a}^{10}{x}^{11}}{11}}-{\frac{{a}^{9}{x}^{10}}{5}}+{\frac{{x}^{9}{a}^{8}}{3}}+{a}^{7}{x}^{8}-{\frac{2\,{x}^{7}{a}^{6}}{7}}-2\,{x}^{6}{a}^{5}-{\frac{2\,{x}^{5}{a}^{4}}{5}}+2\,{x}^{4}{a}^{3}+{x}^{3}{a}^{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^2*c*x^2+c)^5,x)

[Out]

c^5*(-1/11*a^10*x^11-1/5*a^9*x^10+1/3*x^9*a^8+a^7*x^8-2/7*x^7*a^6-2*x^6*a^5-2/5*x^5*a^4+2*x^4*a^3+x^3*a^2-a*x^
2-x)

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Maxima [A]  time = 1.12469, size = 153, normalized size = 1.82 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} 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)^5,x, algorithm="maxima")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x

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Fricas [A]  time = 1.37059, size = 235, normalized size = 2.8 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} 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)^5,x, algorithm="fricas")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x

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Sympy [A]  time = 0.111206, size = 119, normalized size = 1.42 \begin{align*} - \frac{a^{10} c^{5} x^{11}}{11} - \frac{a^{9} c^{5} x^{10}}{5} + \frac{a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac{2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac{2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} 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)**5,x)

[Out]

-a**10*c**5*x**11/11 - a**9*c**5*x**10/5 + a**8*c**5*x**9/3 + a**7*c**5*x**8 - 2*a**6*c**5*x**7/7 - 2*a**5*c**
5*x**6 - 2*a**4*c**5*x**5/5 + 2*a**3*c**5*x**4 + a**2*c**5*x**3 - a*c**5*x**2 - c**5*x

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Giac [A]  time = 1.15025, size = 153, normalized size = 1.82 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} 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)^5,x, algorithm="giac")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x