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

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

[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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Rubi [A]  time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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])

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])

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*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.81 \[ c^2 \left (a^4 (-\log (x))+\frac {2 a^3}{x}-\frac {2 a}{3 x^3}-\frac {1}{4 x^4}\right ) \]

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

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fricas [A]  time = 0.67, size = 42, normalized size = 0.98 \[ -\frac {12 \, a^{4} c^{2} x^{4} \log \relax (x) - 24 \, a^{3} c^{2} x^{3} + 8 \, a c^{2} x + 3 \, c^{2}}{12 \, x^{4}} \]

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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giac [A]  time = 0.23, size = 41, normalized size = 0.95 \[ -a^{4} c^{2} \log \left ({\left | x \right |}\right ) + \frac {24 \, a^{3} c^{2} x^{3} - 8 \, a c^{2} x - 3 \, c^{2}}{12 \, x^{4}} \]

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

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maple [A]  time = 0.03, size = 40, normalized size = 0.93 \[ -\frac {c^{2}}{4 x^{4}}-\frac {2 a \,c^{2}}{3 x^{3}}+\frac {2 a^{3} c^{2}}{x}-a^{4} c^{2} \ln \relax (x ) \]

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 = 0.31, size = 40, normalized size = 0.93 \[ -a^{4} c^{2} \log \relax (x) + \frac {24 \, a^{3} c^{2} x^{3} - 8 \, a c^{2} x - 3 \, c^{2}}{12 \, x^{4}} \]

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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mupad [B]  time = 0.05, size = 32, normalized size = 0.74 \[ -\frac {c^2\,\left (8\,a\,x-24\,a^3\,x^3+12\,a^4\,x^4\,\ln \relax (x)+3\right )}{12\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.22, size = 41, normalized size = 0.95 \[ - a^{4} c^{2} \log {\relax (x )} - \frac {- 24 a^{3} c^{2} x^{3} + 8 a c^{2} x + 3 c^{2}}{12 x^{4}} \]

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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