3.1057 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^4 (c-a^2 c x^2)} \, dx\)

Optimal. Leaf size=71 \[ \frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {3 a^2}{c x}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 71, 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, 44} \[ \frac {a^3}{c (1-a x)}-\frac {3 a^2}{c x}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1}{x^4 (1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{x^4}+\frac {2 a}{x^3}+\frac {3 a^2}{x^2}+\frac {4 a^3}{x}+\frac {a^4}{(-1+a x)^2}-\frac {4 a^4}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {1}{3 c x^3}-\frac {a}{c x^2}-\frac {3 a^2}{c x}+\frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 71, normalized size = 1.00 \[ \frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {3 a^2}{c x}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 83, normalized size = 1.17 \[ -\frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x + 12 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 12 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \relax (x) - 1}{3 \, {\left (a c x^{4} - c x^{3}\right )}} \]

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

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giac [A]  time = 0.31, size = 64, normalized size = 0.90 \[ -\frac {4 \, a^{3} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac {4 \, a^{3} \log \left ({\left | x \right |}\right )}{c} - \frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \, {\left (a x - 1\right )} c x^{3}} \]

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

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maple [A]  time = 0.04, size = 69, normalized size = 0.97 \[ -\frac {1}{3 c \,x^{3}}-\frac {a}{c \,x^{2}}-\frac {3 a^{2}}{c x}+\frac {4 a^{3} \ln \relax (x )}{c}-\frac {a^{3}}{c \left (a x -1\right )}-\frac {4 a^{3} \ln \left (a x -1\right )}{c} \]

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

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maxima [A]  time = 0.31, size = 64, normalized size = 0.90 \[ -\frac {4 \, a^{3} \log \left (a x - 1\right )}{c} + \frac {4 \, a^{3} \log \relax (x)}{c} - \frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \, {\left (a c x^{4} - c x^{3}\right )}} \]

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="maxima")

[Out]

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

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mupad [B]  time = 0.93, size = 55, normalized size = 0.77 \[ \frac {8\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )}{c}-\frac {-4\,a^3\,x^3+2\,a^2\,x^2+\frac {2\,a\,x}{3}+\frac {1}{3}}{c\,x^3-a\,c\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.29, size = 54, normalized size = 0.76 \[ \frac {4 a^{3} \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right )}{c} + \frac {- 12 a^{3} x^{3} + 6 a^{2} x^{2} + 2 a x + 1}{3 a c x^{4} - 3 c x^{3}} \]

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)

[Out]

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

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