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

Optimal. Leaf size=90 \[ -\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{1}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

-Sqrt[1 - a^2*x^2]/(3*x^3) + (a*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (a^3*ArcTanh[Sq
rt[1 - a^2*x^2]])/2

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Rubi [A]  time = 0.082619, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6124, 835, 807, 266, 63, 208} \[ -\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{1}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[1 - a^2*x^2]/(3*x^3) + (a*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (a^3*ArcTanh[Sq
rt[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^4} \, dx &=\int \frac{1-a x}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{1}{3} \int \frac{3 a-2 a^2 x}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}+\frac{1}{6} \int \frac{4 a^2-3 a^3 x}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{1}{2} a^3 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{1}{4} a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{2} a \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}}{3 x^3}+\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0531421, size = 66, normalized size = 0.73 \[ \frac{1}{6} \left (\frac{\left (-4 a^2 x^2+3 a x-2\right ) \sqrt{1-a^2 x^2}}{x^3}+3 a^3 \log \left (\sqrt{1-a^2 x^2}+1\right )-3 a^3 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((-2 + 3*a*x - 4*a^2*x^2)*Sqrt[1 - a^2*x^2])/x^3 - 3*a^3*Log[x] + 3*a^3*Log[1 + Sqrt[1 - a^2*x^2]])/6

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Maple [B]  time = 0.055, size = 207, normalized size = 2.3 \begin{align*} -{\frac{{a}^{2}}{x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{a}^{4}x\sqrt{-{a}^{2}{x}^{2}+1}-{{a}^{4}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{a}^{3}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{{a}^{3}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{a}^{3}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }+{{a}^{4}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{a}{2\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{3\,{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/x*(-a^2*x^2+1)^(3/2)-a^4*x*(-a^2*x^2+1)^(1/2)-a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/
2*a^3*arctanh(1/(-a^2*x^2+1)^(1/2))-1/2*a^3*(-a^2*x^2+1)^(1/2)+a^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+a^4/(a^2
)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))+1/2*a/x^2*(-a^2*x^2+1)^(3/2)-1/3/x^3*(-a^2*x^
2+1)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91862, size = 134, normalized size = 1.49 \begin{align*} -\frac{3 \, a^{3} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (4 \, a^{2} x^{2} - 3 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (4*a^2*x^2 - 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)), x)

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Giac [B]  time = 1.2316, size = 284, normalized size = 3.16 \begin{align*} \frac{{\left (a^{4} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} + \frac{a^{4} \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{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a^4 - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2)*a^6*x^3/((sqrt(
-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) + 1/2*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/ab
s(a) - 1/24*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2/x^2 + (sqrt(-a^
2*x^2 + 1)*abs(a) + a)^3/x^3)/(a^2*abs(a))