3.1056 \(\int \frac{e^{2 \tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)} \, dx\)

Optimal. Leaf size=60 \[ \frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}-\frac{2 a}{c x}-\frac{1}{2 c x^2} \]

[Out]

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

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Rubi [A]  time = 0.101499, antiderivative size = 60, 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} \[ \frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}-\frac{2 a}{c x}-\frac{1}{2 c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*c*x^2) - (2*a)/(c*x) + a^2/(c*(1 - a*x)) + (3*a^2*Log[x])/c - (3*a^2*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^3 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1}{x^3 (1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{x^3}+\frac{2 a}{x^2}+\frac{3 a^2}{x}+\frac{a^3}{(-1+a x)^2}-\frac{3 a^3}{-1+a x}\right ) \, dx}{c}\\ &=-\frac{1}{2 c x^2}-\frac{2 a}{c x}+\frac{a^2}{c (1-a x)}+\frac{3 a^2 \log (x)}{c}-\frac{3 a^2 \log (1-a x)}{c}\\ \end{align*}

Mathematica [A]  time = 0.0477582, size = 52, normalized size = 0.87 \[ \frac{\frac{-6 a^2 x^2+3 a x+1}{x^2 (a x-1)}+6 a^2 \log (x)-6 a^2 \log (1-a x)}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 3*a*x - 6*a^2*x^2)/(x^2*(-1 + a*x)) + 6*a^2*Log[x] - 6*a^2*Log[1 - a*x])/(2*c)

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Maple [A]  time = 0.035, size = 58, normalized size = 1. \begin{align*} -{\frac{1}{2\,c{x}^{2}}}-2\,{\frac{a}{cx}}+3\,{\frac{{a}^{2}\ln \left ( x \right ) }{c}}-{\frac{{a}^{2}}{c \left ( ax-1 \right ) }}-3\,{\frac{{a}^{2}\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^3/(-a^2*c*x^2+c),x)

[Out]

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

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Maxima [A]  time = 0.956291, size = 76, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2} \log \left (a x - 1\right )}{c} + \frac{3 \, a^{2} \log \left (x\right )}{c} - \frac{6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \,{\left (a c x^{3} - c x^{2}\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^3/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.03348, size = 157, normalized size = 2.62 \begin{align*} -\frac{6 \, a^{2} x^{2} - 3 \, a x + 6 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 6 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 1}{2 \,{\left (a c x^{3} - c x^{2}\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^3/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.444763, size = 46, normalized size = 0.77 \begin{align*} \frac{3 a^{2} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right )}{c} - \frac{6 a^{2} x^{2} - 3 a x - 1}{2 a c x^{3} - 2 c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12318, size = 76, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac{3 \, a^{2} \log \left ({\left | x \right |}\right )}{c} - \frac{6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \,{\left (a x - 1\right )} c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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