3.1037 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^2}{x^5} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 a^3 c^2}{x}+a^4 \left (-c^2\right ) \log (x)-\frac{2 a c^2}{3 x^3}-\frac{c^2}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0791424, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 75} \[ \frac{2 a^3 c^2}{x}+a^4 \left (-c^2\right ) \log (x)-\frac{2 a c^2}{3 x^3}-\frac{c^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.0173615, size = 35, normalized size = 0.81 \[ c^2 \left (\frac{2 a^3}{x}+a^4 (-\log (x))-\frac{2 a}{3 x^3}-\frac{1}{4 x^4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^2*(-1/(4*x^4) - (2*a)/(3*x^3) + (2*a^3)/x - a^4*Log[x])

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Maple [A]  time = 0.03, size = 40, normalized size = 0.9 \begin{align*} -{\frac{{c}^{2}}{4\,{x}^{4}}}-{\frac{2\,a{c}^{2}}{3\,{x}^{3}}}+2\,{\frac{{a}^{3}{c}^{2}}{x}}-{a}^{4}{c}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 0.954989, size = 54, normalized size = 1.26 \begin{align*} -a^{4} c^{2} \log \left (x\right ) + \frac{24 \, a^{3} c^{2} x^{3} - 8 \, a c^{2} x - 3 \, c^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^5,x, algorithm="maxima")

[Out]

-a^4*c^2*log(x) + 1/12*(24*a^3*c^2*x^3 - 8*a*c^2*x - 3*c^2)/x^4

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Fricas [A]  time = 1.66952, size = 97, normalized size = 2.26 \begin{align*} -\frac{12 \, a^{4} c^{2} x^{4} \log \left (x\right ) - 24 \, a^{3} c^{2} x^{3} + 8 \, a c^{2} x + 3 \, c^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^5,x, algorithm="fricas")

[Out]

-1/12*(12*a^4*c^2*x^4*log(x) - 24*a^3*c^2*x^3 + 8*a*c^2*x + 3*c^2)/x^4

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Sympy [A]  time = 0.396636, size = 39, normalized size = 0.91 \begin{align*} - a^{4} c^{2} \log{\left (x \right )} + \frac{24 a^{3} c^{2} x^{3} - 8 a c^{2} x - 3 c^{2}}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**5,x)

[Out]

-a**4*c**2*log(x) + (24*a**3*c**2*x**3 - 8*a*c**2*x - 3*c**2)/(12*x**4)

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Giac [A]  time = 1.12034, size = 55, normalized size = 1.28 \begin{align*} -a^{4} c^{2} \log \left ({\left | x \right |}\right ) + \frac{24 \, a^{3} c^{2} x^{3} - 8 \, a c^{2} x - 3 \, c^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^5,x, algorithm="giac")

[Out]

-a^4*c^2*log(abs(x)) + 1/12*(24*a^3*c^2*x^3 - 8*a*c^2*x - 3*c^2)/x^4