3.1071 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1}{16 a^2 c^3 (1-a x)}+\frac{1}{16 a^2 c^3 (a x+1)}+\frac{1}{12 a^2 c^3 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a^2 c^3} \]

[Out]

1/(12*a^2*c^3*(1 - a*x)^3) - 1/(16*a^2*c^3*(1 - a*x)) + 1/(16*a^2*c^3*(1 + a*x)) - ArcTanh[a*x]/(8*a^2*c^3)

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Rubi [A]  time = 0.0880824, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6150, 77, 207} \[ -\frac{1}{16 a^2 c^3 (1-a x)}+\frac{1}{16 a^2 c^3 (a x+1)}+\frac{1}{12 a^2 c^3 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a^2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(12*a^2*c^3*(1 - a*x)^3) - 1/(16*a^2*c^3*(1 - a*x)) + 1/(16*a^2*c^3*(1 + a*x)) - ArcTanh[a*x]/(8*a^2*c^3)

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{4 a (-1+a x)^4}-\frac{1}{16 a (-1+a x)^2}-\frac{1}{16 a (1+a x)^2}+\frac{1}{8 a \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=\frac{1}{12 a^2 c^3 (1-a x)^3}-\frac{1}{16 a^2 c^3 (1-a x)}+\frac{1}{16 a^2 c^3 (1+a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{8 a c^3}\\ &=\frac{1}{12 a^2 c^3 (1-a x)^3}-\frac{1}{16 a^2 c^3 (1-a x)}+\frac{1}{16 a^2 c^3 (1+a x)}-\frac{\tanh ^{-1}(a x)}{8 a^2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0339208, size = 60, normalized size = 0.88 \[ \frac{-\frac{1}{16 a^2 (1-a x)}+\frac{1}{16 a^2 (a x+1)}+\frac{1}{12 a^2 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a^2}}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1/(12*a^2*(1 - a*x)^3) - 1/(16*a^2*(1 - a*x)) + 1/(16*a^2*(1 + a*x)) - ArcTanh[a*x]/(8*a^2))/c^3

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Maple [A]  time = 0.036, size = 75, normalized size = 1.1 \begin{align*}{\frac{1}{16\,{a}^{2}{c}^{3} \left ( ax+1 \right ) }}-{\frac{\ln \left ( ax+1 \right ) }{16\,{a}^{2}{c}^{3}}}-{\frac{1}{12\,{a}^{2}{c}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{1}{16\,{a}^{2}{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{16\,{a}^{2}{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/a^2/c^3/(a*x+1)-1/16/c^3/a^2*ln(a*x+1)-1/12/c^3/a^2/(a*x-1)^3+1/16/c^3/a^2/(a*x-1)+1/16/c^3/a^2*ln(a*x-1)

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Maxima [A]  time = 0.967842, size = 126, normalized size = 1.85 \begin{align*} \frac{3 \, a^{3} x^{3} - 6 \, a^{2} x^{2} + a x - 2}{24 \,{\left (a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{3} c^{3} x - a^{2} c^{3}\right )}} - \frac{\log \left (a x + 1\right )}{16 \, a^{2} c^{3}} + \frac{\log \left (a x - 1\right )}{16 \, a^{2} c^{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/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/24*(3*a^3*x^3 - 6*a^2*x^2 + a*x - 2)/(a^6*c^3*x^4 - 2*a^5*c^3*x^3 + 2*a^3*c^3*x - a^2*c^3) - 1/16*log(a*x +
1)/(a^2*c^3) + 1/16*log(a*x - 1)/(a^2*c^3)

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Fricas [B]  time = 2.31226, size = 269, normalized size = 3.96 \begin{align*} \frac{6 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 2 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 4}{48 \,{\left (a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{3} c^{3} x - a^{2} c^{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/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.640087, size = 87, normalized size = 1.28 \begin{align*} \frac{3 a^{3} x^{3} - 6 a^{2} x^{2} + a x - 2}{24 a^{6} c^{3} x^{4} - 48 a^{5} c^{3} x^{3} + 48 a^{3} c^{3} x - 24 a^{2} c^{3}} + \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{16} - \frac{\log{\left (x + \frac{1}{a} \right )}}{16}}{a^{2} c^{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/(-a**2*c*x**2+c)**3,x)

[Out]

(3*a**3*x**3 - 6*a**2*x**2 + a*x - 2)/(24*a**6*c**3*x**4 - 48*a**5*c**3*x**3 + 48*a**3*c**3*x - 24*a**2*c**3)
+ (log(x - 1/a)/16 - log(x + 1/a)/16)/(a**2*c**3)

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Giac [A]  time = 1.1233, size = 100, normalized size = 1.47 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{2} c^{3}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{2} c^{3}} + \frac{3 \, a^{3} x^{3} - 6 \, a^{2} x^{2} + a x - 2}{24 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} a^{2} c^{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/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/16*log(abs(a*x + 1))/(a^2*c^3) + 1/16*log(abs(a*x - 1))/(a^2*c^3) + 1/24*(3*a^3*x^3 - 6*a^2*x^2 + a*x - 2)/
((a*x + 1)*(a*x - 1)^3*a^2*c^3)