3.1049 \(\int \frac {e^{2 \tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=63 \[ \frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c}+\frac {3 x}{a^4 c}+\frac {x^2}{a^3 c}+\frac {x^3}{3 a^2 c} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 63, 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, 43} \[ \frac {x^3}{3 a^2 c}+\frac {x^2}{a^3 c}+\frac {3 x}{a^4 c}+\frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2),x]

[Out]

(3*x)/(a^4*c) + x^2/(a^3*c) + x^3/(3*a^2*c) + 1/(a^5*c*(1 - a*x)) + (4*Log[1 - a*x])/(a^5*c)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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)} x^4}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^4}{(1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {3}{a^4}+\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {1}{a^4 (-1+a x)^2}+\frac {4}{a^4 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {3 x}{a^4 c}+\frac {x^2}{a^3 c}+\frac {x^3}{3 a^2 c}+\frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 49, normalized size = 0.78 \[ \frac {a^3 x^3+3 a^2 x^2+9 a x+\frac {3}{1-a x}+12 \log (1-a x)}{3 a^5 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2),x]

[Out]

(9*a*x + 3*a^2*x^2 + a^3*x^3 + 3/(1 - a*x) + 12*Log[1 - a*x])/(3*a^5*c)

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fricas [A]  time = 0.54, size = 59, normalized size = 0.94 \[ \frac {a^{4} x^{4} + 2 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 9 \, a x + 12 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 3}{3 \, {\left (a^{6} c x - a^{5} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

1/3*(a^4*x^4 + 2*a^3*x^3 + 6*a^2*x^2 - 9*a*x + 12*(a*x - 1)*log(a*x - 1) - 3)/(a^6*c*x - a^5*c)

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giac [A]  time = 0.18, size = 70, normalized size = 1.11 \[ \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{5} c} - \frac {1}{{\left (a x - 1\right )} a^{5} c} + \frac {a^{4} c^{2} x^{3} + 3 \, a^{3} c^{2} x^{2} + 9 \, a^{2} c^{2} x}{3 \, a^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

4*log(abs(a*x - 1))/(a^5*c) - 1/((a*x - 1)*a^5*c) + 1/3*(a^4*c^2*x^3 + 3*a^3*c^2*x^2 + 9*a^2*c^2*x)/(a^6*c^3)

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maple [A]  time = 0.03, size = 61, normalized size = 0.97 \[ \frac {x^{3}}{3 a^{2} c}+\frac {x^{2}}{a^{3} c}+\frac {3 x}{a^{4} c}-\frac {1}{c \,a^{5} \left (a x -1\right )}+\frac {4 \ln \left (a x -1\right )}{c \,a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c),x)

[Out]

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

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maxima [A]  time = 0.30, size = 57, normalized size = 0.90 \[ -\frac {1}{a^{6} c x - a^{5} c} + \frac {a^{2} x^{3} + 3 \, a x^{2} + 9 \, x}{3 \, a^{4} c} + \frac {4 \, \log \left (a x - 1\right )}{a^{5} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

-1/(a^6*c*x - a^5*c) + 1/3*(a^2*x^3 + 3*a*x^2 + 9*x)/(a^4*c) + 4*log(a*x - 1)/(a^5*c)

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mupad [B]  time = 0.05, size = 64, normalized size = 1.02 \[ \frac {1}{a\,\left (a^4\,c-a^5\,c\,x\right )}+\frac {3\,x}{a^4\,c}+\frac {x^3}{3\,a^2\,c}+\frac {x^2}{a^3\,c}+\frac {4\,\ln \left (a\,x-1\right )}{a^5\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(a*x + 1)^2)/((c - a^2*c*x^2)*(a^2*x^2 - 1)),x)

[Out]

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

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sympy [A]  time = 0.19, size = 53, normalized size = 0.84 \[ - \frac {1}{a^{6} c x - a^{5} c} + \frac {x^{3}}{3 a^{2} c} + \frac {x^{2}}{a^{3} c} + \frac {3 x}{a^{4} c} + \frac {4 \log {\left (a x - 1 \right )}}{a^{5} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**4/(-a**2*c*x**2+c),x)

[Out]

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

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