### 3.782 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^3 \, dx$$

Optimal. Leaf size=76 $\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}-\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 x$

[Out]

-c^3/(5*a^6*x^5) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*Log[x]
)/a

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Rubi [A]  time = 0.159287, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6167, 6157, 6150, 88} $\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}-\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-c^3/(5*a^6*x^5) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*Log[x]
)/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 6157

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 - a^2*x^
2)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx\\ &=\frac{c^3 \int \frac{e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=\frac{c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x^6} \, dx}{a^6}\\ &=\frac{c^3 \int \left (a^6+\frac{1}{x^6}+\frac{2 a}{x^5}-\frac{a^2}{x^4}-\frac{4 a^3}{x^3}-\frac{a^4}{x^2}+\frac{2 a^5}{x}\right ) \, dx}{a^6}\\ &=-\frac{c^3}{5 a^6 x^5}-\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}+\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}+c^3 x+\frac{2 c^3 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0265082, size = 76, normalized size = 1. $\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}-\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-c^3/(5*a^6*x^5) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*Log[x]
)/a

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Maple [A]  time = 0.047, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{5\,{a}^{6}{x}^{5}}}-{\frac{{c}^{3}}{2\,{a}^{5}{x}^{4}}}+{\frac{{c}^{3}}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{{c}^{3}}{{x}^{2}{a}^{3}}}+{\frac{{c}^{3}}{{a}^{2}x}}+{c}^{3}x+2\,{\frac{{c}^{3}\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00869, size = 95, normalized size = 1.25 \begin{align*} c^{3} x + \frac{2 \, c^{3} \log \left (x\right )}{a} + \frac{30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*x + 2*c^3*log(x)/a + 1/30*(30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^6*x^5
)

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Fricas [A]  time = 1.10598, size = 174, normalized size = 2.29 \begin{align*} \frac{30 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(30*a^6*c^3*x^6 + 60*a^5*c^3*x^5*log(x) + 30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 - 15*a*c^3*x -
6*c^3)/(a^6*x^5)

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Sympy [A]  time = 0.510331, size = 76, normalized size = 1. \begin{align*} \frac{a^{6} c^{3} x + 2 a^{5} c^{3} \log{\left (x \right )} + \frac{30 a^{4} c^{3} x^{4} + 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} - 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**6*c**3*x + 2*a**5*c**3*log(x) + (30*a**4*c**3*x**4 + 60*a**3*c**3*x**3 + 10*a**2*c**3*x**2 - 15*a*c**3*x -
6*c**3)/(30*x**5))/a**6

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Giac [A]  time = 1.12859, size = 96, normalized size = 1.26 \begin{align*} c^{3} x + \frac{2 \, c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac{30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*x + 2*c^3*log(abs(x))/a + 1/30*(30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^
6*x^5)