\(\int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx\) [102]

Optimal result

Integrand size = 16, antiderivative size = 92 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}+3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]

[Out]

-(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x+3*I*a*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))+3*I*a*arctanh((1+I*a*x)^(1/4)
/(1-I*a*x)^(1/4))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5170, 96, 95, 218, 212, 209} \[ \int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x} \]

[In]

Int[1/(E^(((3*I)/2)*ArcTan[a*x])*x^2),x]

[Out]

-(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x) + (3*I)*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (3*I)*a*Arc
Tanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 209

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 5170

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^{3/4}}{x^2 (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-\frac {1}{2} (3 i a) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}-(6 i a) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}+(3 i a) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+(3 i a) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}+3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\frac {i (1-i a x)^{3/4} \left (i-a x+2 a x \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )\right )}{x (1+i a x)^{3/4}} \]

[In]

Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x^2),x]

[Out]

(I*(1 - I*a*x)^(3/4)*(I - a*x + 2*a*x*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(3/
4))

Maple [F]

\[\int \frac {1}{{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {3}{2}} x^{2}}d x\]

[In]

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

[Out]

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

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. 152 vs. \(2 (64) = 128\).

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\frac {3 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 3 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 3 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 3 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (-i \, a x + 1\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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