3.10 \(\int \frac{e^{i \tan ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=113 \[ \frac{2 i a^3 \sqrt{a^2 x^2+1}}{3 x}+\frac{3 a^2 \sqrt{a^2 x^2+1}}{8 x^2}-\frac{i a \sqrt{a^2 x^2+1}}{3 x^3}-\frac{\sqrt{a^2 x^2+1}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0904184, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5060, 835, 807, 266, 63, 208} \[ \frac{2 i a^3 \sqrt{a^2 x^2+1}}{3 x}+\frac{3 a^2 \sqrt{a^2 x^2+1}}{8 x^2}-\frac{i a \sqrt{a^2 x^2+1}}{3 x^3}-\frac{\sqrt{a^2 x^2+1}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(I*ArcTan[a*x])/x^5,x]

[Out]

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

Rule 5060

Int[E^(ArcTan[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n
- 1)/2)*Sqrt[1 + a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(I*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^{i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac{1+i 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 i 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{i a \sqrt{1+a^2 x^2}}{3 x^3}+\frac{1}{12} \int \frac{-9 a^2-8 i a^3 x}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{4 x^4}-\frac{i 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 i 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{i 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 i 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{i 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 i 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{i 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 i 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{i 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 i 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 = 0.0574363, size = 76, normalized size = 0.67 \[ \frac{1}{24} \left (\frac{\sqrt{a^2 x^2+1} \left (16 i a^3 x^3+9 a^2 x^2-8 i a x-6\right )}{x^4}-9 a^4 \log \left (\sqrt{a^2 x^2+1}+1\right )+9 a^4 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(I*ArcTan[a*x])/x^5,x]

[Out]

((Sqrt[1 + a^2*x^2]*(-6 - (8*I)*a*x + 9*a^2*x^2 + (16*I)*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 [A]  time = 0.07, size = 97, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{3\,{a}^{2}}{4} \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 ) }+ia \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{2\,{a}^{2}}{3\,x}\sqrt{{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)/(a^2*x^2+1)^(1/2)/x^5,x)

[Out]

-1/4*(a^2*x^2+1)^(1/2)/x^4-3/4*a^2*(-1/2*(a^2*x^2+1)^(1/2)/x^2+1/2*a^2*arctanh(1/(a^2*x^2+1)^(1/2)))+I*a*(-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 = 1.04937, size = 119, normalized size = 1.05 \begin{align*} -\frac{3}{8} \, a^{4} \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) + \frac{2 i \, \sqrt{a^{2} x^{2} + 1} a^{3}}{3 \, x} + \frac{3 \, \sqrt{a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac{i \, \sqrt{a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac{\sqrt{a^{2} x^{2} + 1}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)/(a^2*x^2+1)^(1/2)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72206, size = 242, normalized size = 2.14 \begin{align*} -\frac{9 \, a^{4} x^{4} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - 9 \, a^{4} x^{4} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) - 16 i \, a^{4} x^{4} -{\left (16 i \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 8 i \, 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+I*a*x)/(a^2*x^2+1)^(1/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.44941, size = 122, normalized size = 1.08 \begin{align*} \frac{2 i a^{4} \sqrt{1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{3 a^{4} \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{8} + \frac{3 a^{3}}{8 x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{i a^{2} \sqrt{1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} + \frac{a}{8 x^{3} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)/(a**2*x**2+1)**(1/2)/x**5,x)

[Out]

2*I*a**4*sqrt(1 + 1/(a**2*x**2))/3 - 3*a**4*asinh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(1 + 1/(a**2*x**2))) - I*a**2*s
qrt(1 + 1/(a**2*x**2))/(3*x**2) + a/(8*x**3*sqrt(1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(1 + 1/(a**2*x**2)))

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Giac [B]  time = 1.15453, size = 324, normalized size = 2.87 \begin{align*} -\frac{3}{8} \, a^{4} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} + 1 \right |}\right ) + \frac{3}{8} \, a^{4} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1} - 1 \right |}\right ) - \frac{9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{7} a^{4} - 33 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{5} a^{4} - 48 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{4} a^{3} i{\left | a \right |} - 33 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{3} a^{4} + 64 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} a^{3} i{\left | a \right |} + 9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )} a^{4} - 16 \, a^{3} i{\left | a \right |}}{12 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)/(a^2*x^2+1)^(1/2)/x^5,x, algorithm="giac")

[Out]

-3/8*a^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 + 1) + 1)) + 3/8*a^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 + 1) - 1)) - 1
/12*(9*(x*abs(a) - sqrt(a^2*x^2 + 1))^7*a^4 - 33*(x*abs(a) - sqrt(a^2*x^2 + 1))^5*a^4 - 48*(x*abs(a) - sqrt(a^
2*x^2 + 1))^4*a^3*i*abs(a) - 33*(x*abs(a) - sqrt(a^2*x^2 + 1))^3*a^4 + 64*(x*abs(a) - sqrt(a^2*x^2 + 1))^2*a^3
*i*abs(a) + 9*(x*abs(a) - sqrt(a^2*x^2 + 1))*a^4 - 16*a^3*i*abs(a))/((x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)^4