\(\int \frac {e^{-3 i \arctan (a x)}}{x} \, dx\) [56]

Optimal result

Integrand size = 14, antiderivative size = 52 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]

[Out]

I*arcsinh(a*x)-arctanh((a^2*x^2+1)^(1/2))+4*I*(a^2*x^2+1)^(1/2)/(I-a*x)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5168, 6874, 221, 272, 65, 214, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 i \sqrt {a^2 x^2+1}}{-a x+i}+i \text {arcsinh}(a x) \]

[In]

Int[1/(E^((3*I)*ArcTan[a*x])*x),x]

[Out]

((4*I)*Sqrt[1 + a^2*x^2])/(I - a*x) + 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 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^2}{x (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {i a}{\sqrt {1+a^2 x^2}}+\frac {1}{x \sqrt {1+a^2 x^2}}-\frac {4 a}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = (i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx-(4 a) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+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 ) \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+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} \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\frac {4 i \sqrt {1+a^2 x^2}}{-i+a x}+i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \]

[In]

Integrate[1/(E^((3*I)*ArcTan[a*x])*x),x]

[Out]

((-4*I)*Sqrt[1 + a^2*x^2])/(-I + a*x) + 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. 648 vs. \(2 (45 ) = 90\).

Time = 0.29 (sec) , antiderivative size = 649, normalized size of antiderivative = 12.48

method	result	size
default	\(\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}-\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}-i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )+\frac {i \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a}\)	\(649\)

[In]

int(1/(1+I*a*x)^3*(a^2*x^2+1)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

1/3*(a^2*x^2+1)^(3/2)+(a^2*x^2+1)^(1/2)-arctanh(1/(a^2*x^2+1)^(1/2))+1/a^2*(I/a/(x-I/a)^3*((x-I/a)^2*a^2+2*I*a
*(x-I/a))^(5/2)-2*I*a*(-I/a/(x-I/a)^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(5/2)+3*I*a*(1/3*((x-I/a)^2*a^2+2*I*a*(x-I
/a))^(3/2)+I*a*(1/4*(2*(x-I/a)*a^2+2*I*a)/a^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(1/2)+1/2*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)))))-1/3*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(3/2)-I*a*(1/4
*(2*(x-I/a)*a^2+2*I*a)/a^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(1/2)+1/2*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))+I/a*(-I/a/(x-I/a)^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(5/2)+3*I*a*(1/3*((x
-I/a)^2*a^2+2*I*a*(x-I/a))^(3/2)+I*a*(1/4*(2*(x-I/a)*a^2+2*I*a)/a^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(1/2)+1/2*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. 100 vs. \(2 (41) = 82\).

Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.92 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {-4 i \, a x - {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x - i} \]

[In]

integrate(1/(1+I*a*x)^3*(a^2*x^2+1)^(3/2)/x,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx\right ) \]

[In]

integrate(1/(1+I*a*x)**3*(a**2*x**2+1)**(3/2)/x,x)

[Out]

I*(Integral(sqrt(a**2*x**2 + 1)/(a**3*x**4 - 3*I*a**2*x**3 - 3*a*x**2 + I*x), x) + Integral(a**2*x**2*sqrt(a**
2*x**2 + 1)/(a**3*x**4 - 3*I*a**2*x**3 - 3*a*x**2 + I*x), x))

Maxima [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x} \,d x } \]

[In]

integrate(1/(1+I*a*x)^3*(a^2*x^2+1)^(3/2)/x,x, algorithm="maxima")

[Out]

integrate((a^2*x^2 + 1)^(3/2)/((I*a*x + 1)^3*x), x)

Giac [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x} \,d x } \]

[In]

integrate(1/(1+I*a*x)^3*(a^2*x^2+1)^(3/2)/x,x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]

[In]

int((a^2*x^2 + 1)^(3/2)/(x*(a*x*1i + 1)^3),x)

[Out]

(a*asinh(x*(a^2)^(1/2))*1i)/(a^2)^(1/2) - atanh((a^2*x^2 + 1)^(1/2)) + (a*(a^2*x^2 + 1)^(1/2)*4i)/((((a^2)^(1/
2)*1i)/a - x*(a^2)^(1/2))*(a^2)^(1/2))