3.42 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

-Sqrt[1 - a^2*x^2]/(4*x^4) + (a*Sqrt[1 - a^2*x^2])/(3*x^3) - (3*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (2*a^3*Sqrt[1
- a^2*x^2])/(3*x) - (3*a^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

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Rubi [A]  time = 0.103138, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*x^5),x]

[Out]

-Sqrt[1 - a^2*x^2]/(4*x^4) + (a*Sqrt[1 - a^2*x^2])/(3*x^3) - (3*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (2*a^3*Sqrt[1
- a^2*x^2])/(3*x) - (3*a^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

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^5} \, dx &=\int \frac{1-a x}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{1}{4} \int \frac{4 a-3 a^2 x}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}+\frac{1}{12} \int \frac{9 a^2-8 a^3 x}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{1}{24} \int \frac{16 a^3-9 a^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{8} \left (3 a^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{16} \left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{1}{8} \left (3 a^2\right ) \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}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.057365, size = 74, normalized size = 0.65 \[ \frac{1}{24} \left (\frac{\sqrt{1-a^2 x^2} \left (16 a^3 x^3-9 a^2 x^2+8 a x-6\right )}{x^4}-9 a^4 \log \left (\sqrt{1-a^2 x^2}+1\right )+9 a^4 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcTanh[a*x]*x^5),x]

[Out]

((Sqrt[1 - a^2*x^2]*(-6 + 8*a*x - 9*a^2*x^2 + 16*a^3*x^3))/x^4 + 9*a^4*Log[x] - 9*a^4*Log[1 + Sqrt[1 - a^2*x^2
]])/24

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Maple [B]  time = 0.054, size = 226, normalized size = 2. \begin{align*} -{\frac{1}{4\,{x}^{4}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}}{8\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{4}}{8}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,{a}^{4}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{a}^{3}}{x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{a}^{5}x\sqrt{-{a}^{2}{x}^{2}+1}+{{a}^{5}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{a}^{4}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }-{{a}^{5}\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}{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^5,x)

[Out]

-1/4/x^4*(-a^2*x^2+1)^(3/2)-5/8*a^2/x^2*(-a^2*x^2+1)^(3/2)-3/8*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))+3/8*a^4*(-a^2
*x^2+1)^(1/2)+a^3/x*(-a^2*x^2+1)^(3/2)+a^5*x*(-a^2*x^2+1)^(1/2)+a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2
+1)^(1/2))-a^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+
1/a))^(1/2))+1/3*a/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^{5}}\,{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^5,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*x^5), x)

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Fricas [A]  time = 1.98605, size = 151, normalized size = 1.32 \begin{align*} \frac{9 \, a^{4} x^{4} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (16 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 8 \, a x - 6\right )} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/24*(9*a^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (16*a^3*x^3 - 9*a^2*x^2 + 8*a*x - 6)*sqrt(-a^2*x^2 + 1))/x^4

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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^{5} \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**5,x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**5*(a*x + 1)), x)

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Giac [B]  time = 1.20479, size = 369, normalized size = 3.24 \begin{align*} \frac{{\left (3 \, a^{5} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{3}}{x} + \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac{72 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}} - \frac{3 \, a^{5} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} + \frac{\frac{72 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5}{\left | a \right |}}{x} - \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3}{\left | a \right |}}{x^{2}} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a{\left | a \right |}}{x^{3}} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \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^5,x, algorithm="giac")

[Out]

1/192*(3*a^5 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a/x^2 - 72*(sqrt
(-a^2*x^2 + 1)*abs(a) + a)^3/(a*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)) - 3/8*a^5*log(1/2*abs
(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/192*(72*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*abs(
a)/x - 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*abs(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*abs(a)/x^3
- 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)/(a*x^4))/a^4