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

Optimal. Leaf size=42 \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.090742, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 44} \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^4*Sqrt[1 - a^2*x^2]),x]

[Out]

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

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^{\tanh ^{-1}(a x)}}{x^4 \sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{x^4 (1-a x)} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{a}{x^3}+\frac{a^2}{x^2}+\frac{a^3}{x}-\frac{a^4}{-1+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{a}{2 x^2}-\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)\\ \end{align*}

Mathematica [A]  time = 0.015382, size = 42, normalized size = 1. \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^4*Sqrt[1 - a^2*x^2]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.031, size = 38, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{\frac{a}{2\,{x}^{2}}}-{\frac{{a}^{2}}{x}}+{a}^{3}\ln \left ( x \right ) -{a}^{3}\ln \left ( ax-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.957407, size = 50, normalized size = 1.19 \begin{align*} -a^{3} \log \left (a x - 1\right ) + a^{3} \log \left (x\right ) - \frac{6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.85445, size = 105, normalized size = 2.5 \begin{align*} -\frac{6 \, a^{3} x^{3} \log \left (a x - 1\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) + 6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.340384, size = 34, normalized size = 0.81 \begin{align*} - a^{3} \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{6 a^{2} x^{2} + 3 a x + 2}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13409, size = 53, normalized size = 1.26 \begin{align*} -a^{3} \log \left ({\left | a x - 1 \right |}\right ) + a^{3} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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