\(\int \frac {e^{-i \arctan (a x)}}{x} \, dx\) [39]

Optimal result

Integrand size = 14, antiderivative size = 25 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5168, 858, 221, 272, 65, 214} \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-i \text {arcsinh}(a x) \]

[In]

Int[1/(E^(I*ArcTan[a*x])*x),x]

[Out]

(-I)*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]

Rule 65

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 214

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 272

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 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 5168

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1-i a x}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\left ((i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\right )+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -i \text {arcsinh}(a x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -i \text {arcsinh}(a x)+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a^2} \\ & = -i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \]

[In]

Integrate[1/(E^(I*ArcTan[a*x])*x),x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22 ) = 44\).

Time = 0.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.84

method	result	size
default	\(\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}-\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\)	\(121\)

[In]

int(1/(1+I*a*x)*(a^2*x^2+1)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-\log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=- i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{2} - i x}\, dx \]

[In]

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

[Out]

-I*Integral(sqrt(a**2*x**2 + 1)/(a*x**2 - I*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-i \, a {\left (\frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {i \, \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right )}{a}\right )} \]

[In]

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

[Out]

-I*a*(arcsinh(a*x)/a - I*arcsinh(1/(a*abs(x)))/a)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=\frac {i \, a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}} \]

[In]

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

[Out]

- atanh((a^2*x^2 + 1)^(1/2)) - (a*asinh(x*(a^2)^(1/2))*1i)/(a^2)^(1/2)