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

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 = 0.0851134, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{1}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[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 = 0.0523851, size = 67, normalized size = 0.74 \[ \frac{1}{6} \left (-\frac{\sqrt{1-a^2 x^2} \left (4 a^2 x^2+3 a x+2\right )}{x^3}-3 a^3 \log \left (\sqrt{1-a^2 x^2}+1\right )+3 a^3 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-((Sqrt[1 - a^2*x^2]*(2 + 3*a*x + 4*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 [A]  time = 0.037, size = 77, normalized size = 0.9 \begin{align*} a \left ( -{\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 ) } \right ) -{\frac{1}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}} \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^4,x)

[Out]

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

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Maxima [A]  time = 1.42523, size = 117, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, a^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{3 \, x} - \frac{\sqrt{-a^{2} x^{2} + 1} a}{2 \, x^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.60205, size = 132, normalized size = 1.47 \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((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 [C]  time = 6.05506, size = 185, normalized size = 2.06 \begin{align*} a \left (\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}\right ) + \begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \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**4,x)

[Out]

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

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Giac [B]  time = 1.22321, 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((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))