3.1046 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x^3} \, dx\)

Optimal. Leaf size=78 \[ \frac {1}{4} a^6 c^3 x^4+\frac {2}{3} a^5 c^3 x^3-\frac {1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac {2 a c^3}{x}-\frac {c^3}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 78, 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, 88} \[ \frac {1}{4} a^6 c^3 x^4+\frac {2}{3} a^5 c^3 x^3-\frac {1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac {2 a c^3}{x}-\frac {c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

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

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Mathematica [A]  time = 0.02, size = 58, normalized size = 0.74 \[ \frac {c^3 \left (3 a^6 x^6+8 a^5 x^5-6 a^4 x^4-48 a^3 x^3-12 a^2 x^2 \log (x)-24 a x-6\right )}{12 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-6 - 24*a*x - 48*a^3*x^3 - 6*a^4*x^4 + 8*a^5*x^5 + 3*a^6*x^6 - 12*a^2*x^2*Log[x]))/(12*x^2)

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fricas [A]  time = 0.67, size = 75, normalized size = 0.96 \[ \frac {3 \, a^{6} c^{3} x^{6} + 8 \, a^{5} c^{3} x^{5} - 6 \, a^{4} c^{3} x^{4} - 48 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} \log \relax (x) - 24 \, a c^{3} x - 6 \, c^{3}}{12 \, x^{2}} \]

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)^3/x^3,x, algorithm="fricas")

[Out]

1/12*(3*a^6*c^3*x^6 + 8*a^5*c^3*x^5 - 6*a^4*c^3*x^4 - 48*a^3*c^3*x^3 - 12*a^2*c^3*x^2*log(x) - 24*a*c^3*x - 6*
c^3)/x^2

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giac [A]  time = 1.04, size = 70, normalized size = 0.90 \[ \frac {1}{4} \, a^{6} c^{3} x^{4} + \frac {2}{3} \, a^{5} c^{3} x^{3} - \frac {1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left ({\left | x \right |}\right ) - \frac {4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \]

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)^3/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 71, normalized size = 0.91 \[ -\frac {c^{3}}{2 x^{2}}-\frac {2 a \,c^{3}}{x}-4 a^{3} c^{3} x -\frac {a^{4} c^{3} x^{2}}{2}+\frac {2 a^{5} c^{3} x^{3}}{3}+\frac {a^{6} c^{3} x^{4}}{4}-a^{2} c^{3} \ln \relax (x ) \]

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)^3/x^3,x)

[Out]

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

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maxima [A]  time = 0.31, size = 69, normalized size = 0.88 \[ \frac {1}{4} \, a^{6} c^{3} x^{4} + \frac {2}{3} \, a^{5} c^{3} x^{3} - \frac {1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \relax (x) - \frac {4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \]

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)^3/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 71, normalized size = 0.91 \[ \frac {2\,a^5\,c^3\,x^3}{3}-4\,a^3\,c^3\,x-\frac {a^4\,c^3\,x^2}{2}-\frac {\frac {c^3}{2}+2\,a\,c^3\,x}{x^2}+\frac {a^6\,c^3\,x^4}{4}-a^2\,c^3\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.20, size = 75, normalized size = 0.96 \[ \frac {a^{6} c^{3} x^{4}}{4} + \frac {2 a^{5} c^{3} x^{3}}{3} - \frac {a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log {\relax (x )} + \frac {- 4 a c^{3} x - c^{3}}{2 x^{2}} \]

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)**3/x**3,x)

[Out]

a**6*c**3*x**4/4 + 2*a**5*c**3*x**3/3 - a**4*c**3*x**2/2 - 4*a**3*c**3*x - a**2*c**3*log(x) + (-4*a*c**3*x - c
**3)/(2*x**2)

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