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

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

[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

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Rubi [A]  time = 0.109847, antiderivative size = 71, 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, 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 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 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])

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*}

Mathematica [A]  time = 0.0393452, size = 71, normalized size = 1. \[ \frac{a^3}{c (1-a x)}-\frac{3 a^2}{c x}+\frac{4 a^3 \log (x)}{c}-\frac{4 a^3 \log (1-a x)}{c}-\frac{a}{c x^2}-\frac{1}{3 c x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[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

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Maple [A]  time = 0.034, size = 69, normalized size = 1. \begin{align*} -{\frac{1}{3\,c{x}^{3}}}-{\frac{a}{c{x}^{2}}}-3\,{\frac{{a}^{2}}{cx}}+4\,{\frac{{a}^{3}\ln \left ( x \right ) }{c}}-{\frac{{a}^{3}}{c \left ( ax-1 \right ) }}-4\,{\frac{{a}^{3}\ln \left ( ax-1 \right ) }{c}} \end{align*}

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 = 0.94526, size = 86, normalized size = 1.21 \begin{align*} -\frac{4 \, a^{3} \log \left (a x - 1\right )}{c} + \frac{4 \, a^{3} \log \left (x\right )}{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 )}} \end{align*}

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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Fricas [A]  time = 2.04781, size = 177, normalized size = 2.49 \begin{align*} -\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 \left (x\right ) - 1}{3 \,{\left (a c x^{4} - c x^{3}\right )}} \end{align*}

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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Sympy [A]  time = 0.492618, size = 54, normalized size = 0.76 \begin{align*} \frac{4 a^{3} \left (\log{\left (x \right )} - \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}} \end{align*}

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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Giac [A]  time = 1.15145, size = 86, normalized size = 1.21 \begin{align*} -\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}} \end{align*}

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)