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

Optimal. Leaf size=64 \[ -\frac{a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

-Sqrt[1 - a^2*x^2]/(2*x^2) - (a*Sqrt[1 - a^2*x^2])/x - (a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2

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Rubi [A]  time = 0.0626748, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6124, 835, 807, 266, 63, 208} \[ -\frac{a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/x^3,x]

[Out]

-Sqrt[1 - a^2*x^2]/(2*x^2) - (a*Sqrt[1 - a^2*x^2])/x - (a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2

Rule 6124

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/
2)*Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0399536, size = 58, normalized size = 0.91 \[ \frac{1}{2} \left (-\frac{(2 a x+1) \sqrt{1-a^2 x^2}}{x^2}-a^2 \log \left (\sqrt{1-a^2 x^2}+1\right )+a^2 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]/x^3,x]

[Out]

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

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Maple [A]  time = 0.037, size = 55, normalized size = 0.9 \begin{align*} -{\frac{a}{x}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(-a^2*x^2+1)^(1/2)/x-1/2*(-a^2*x^2+1)^(1/2)/x^2-1/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.4343, size = 90, normalized size = 1.41 \begin{align*} -\frac{1}{2} \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} a}{x} - \frac{\sqrt{-a^{2} x^{2} + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73341, size = 113, normalized size = 1.77 \begin{align*} \frac{a^{2} x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x + 1\right )}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.39093, size = 136, normalized size = 2.12 \begin{align*} a \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + Piecewise((-a**
2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2
*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))

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Giac [B]  time = 1.2075, size = 213, normalized size = 3.33 \begin{align*} \frac{{\left (a^{3} + \frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}} - \frac{a^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(a^3 + 4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x)*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)) - 1/2*a^3
*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/8*(4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)
*a*abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)/(a*x^2))/a^2