3.671 \(\int e^{-2 \tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^3 \, dx\)

Optimal. Leaf size=78 \[ \frac{2 c^3}{a^3 x^2}-\frac{c^3}{3 a^4 x^3}-\frac{c^3}{2 a^5 x^4}+\frac{c^3}{5 a^6 x^5}-\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 (-x) \]

[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 = 0.128178, antiderivative size = 78, 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 = {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[(c - c/(a^2*x^2))^3/E^(2*ArcTanh[a*x]),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 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 \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)^4 (1+a x)^2}{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 = 0.0217449, size = 78, 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[(c - c/(a^2*x^2))^3/E^(2*ArcTanh[a*x]),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 = 0.037, size = 73, 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((c-c/a^2/x^2)^3/(a*x+1)^2*(-a^2*x^2+1),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 = 0.967265, size = 96, normalized size = 1.23 \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((c-c/a^2/x^2)^3/(a*x+1)^2*(-a^2*x^2+1),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 = 1.85926, size = 176, normalized size = 2.26 \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((c-c/a^2/x^2)^3/(a*x+1)^2*(-a^2*x^2+1),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 = 2.21957, size = 76, normalized size = 0.97 \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((c-c/a**2/x**2)**3/(a*x+1)**2*(-a**2*x**2+1),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 = 1.23353, size = 184, normalized size = 2.36 \begin{align*} -\frac{2 \, c^{3} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{3} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (30 \, c^{3} - \frac{71 \, c^{3}}{a x + 1} - \frac{65 \, c^{3}}{{\left (a x + 1\right )}^{2}} + \frac{310 \, c^{3}}{{\left (a x + 1\right )}^{3}} - \frac{270 \, c^{3}}{{\left (a x + 1\right )}^{4}} + \frac{60 \, c^{3}}{{\left (a x + 1\right )}^{5}}\right )}{\left (a x + 1\right )}}{30 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^3*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a + 2*c^3*log(abs(-1/(a*x + 1) + 1))/a + 1/30*(30*c^3 - 71*c^3/(
a*x + 1) - 65*c^3/(a*x + 1)^2 + 310*c^3/(a*x + 1)^3 - 270*c^3/(a*x + 1)^4 + 60*c^3/(a*x + 1)^5)*(a*x + 1)/(a*(
1/(a*x + 1) - 1)^5)