\(\int \frac {e^{-3 i \arctan (a x)}}{x} \, dx\) [71]

Optimal result

Integrand size = 14, antiderivative size = 48 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {4 (1-i a x)}{\sqrt {1+a^2 x^2}}+i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\frac {4 i \sqrt {1+a^2 x^2}}{-i+a x}+i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \] Input:

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

Output:

((-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]]
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5583, 2351, 564, 25, 243, 73, 221, 671, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5583

\(\displaystyle \int \frac {(1-i a x)^2}{x (1+i a x) \sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 2351

\(\displaystyle \int \frac {1}{x (i a x+1) \sqrt {a^2 x^2+1}}dx+\int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 564

\(\displaystyle -\int -\frac {1}{x \sqrt {a^2 x^2+1}}dx+\int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx+\frac {\sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {1}{x \sqrt {a^2 x^2+1}}dx+\int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx+\frac {\sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \sqrt {a^2 x^2+1}}dx^2+\int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx+\frac {\sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 73

\(\displaystyle \int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx+\frac {\int \frac {1}{\frac {x^4}{a^2}-\frac {1}{a^2}}d\sqrt {a^2 x^2+1}}{a^2}+\frac {\sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 221

\(\displaystyle \int \frac {-x a^2-2 i a}{(i a x+1) \sqrt {a^2 x^2+1}}dx-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {\sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 671

\(\displaystyle i a \int \frac {1}{\sqrt {a^2 x^2+1}}dx-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 \sqrt {a^2 x^2+1}}{1+i a x}\)

\(\Big \downarrow \) 222

\(\displaystyle -\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 \sqrt {a^2 x^2+1}}{1+i a x}+i \text {arcsinh}(a x)\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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 222

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 564

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b 
^(n + 2)*(c + d*x))), x] - Simp[d^(2*n + 2)/b^(n + 1)   Int[(x^m/Sqrt[a + b 
*x^2])*ExpandToSum[((2^(-n - 1)*(-c)^(m - n - 1))/(d^m*x^m) - (-c + d*x)^(- 
n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^ 
2, 0] && ILtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
 

rule 671

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

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

rule 5583

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] /; Free 
Q[{a, m}, x] && IntegerQ[(I*n - 1)/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. \(2 (42 ) = 84\).

Time = 0.19 (sec) , antiderivative size = 649, normalized size of antiderivative = 13.52

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 {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}-\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 )\)	\(649\)

Input:

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

Output:

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)))))+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))
 

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. \(2 (40) = 80\).

Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.08 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {-4 i \, a x - {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x - i} \] Input:

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

Output:

(-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]

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

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

Output:

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]

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

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

Output:

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

Giac [F]

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

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

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 22.88 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \] Input:

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

Output:

(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))
 

Reduce [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {\sqrt {a^{2} x^{2}+1}}{3}-3 \left (\int -\frac {\sqrt {a^{2} x^{2}+1}}{a^{5} x^{5}-3 a^{4} i \,x^{4}-2 a^{3} x^{3}-2 a^{2} i \,x^{2}-3 a x +i}d x \right ) a +\frac {\left (\int -\frac {\sqrt {a^{2} x^{2}+1}\, x^{4}}{a^{5} x^{5}-3 a^{4} i \,x^{4}-2 a^{3} x^{3}-2 a^{2} i \,x^{2}-3 a x +i}d x \right ) a^{5}}{3}-\left (\int -\frac {\sqrt {a^{2} x^{2}+1}\, x^{3}}{a^{5} x^{5}-3 a^{4} i \,x^{4}-2 a^{3} x^{3}-2 a^{2} i \,x^{2}-3 a x +i}d x \right ) a^{4} i -\frac {11 \left (\int -\frac {\sqrt {a^{2} x^{2}+1}\, x}{a^{5} x^{5}-3 a^{4} i \,x^{4}-2 a^{3} x^{3}-2 a^{2} i \,x^{2}-3 a x +i}d x \right ) a^{2} i}{3}-\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, x}{a^{3} i \,x^{3}+3 a^{2} x^{2}-3 a i x -1}d x \right ) a^{2}+\frac {\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}-1\right )}{2}-\frac {\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+1\right )}{2} \] Input:

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

Output:

(2*sqrt(a**2*x**2 + 1) - 18*int(( - sqrt(a**2*x**2 + 1))/(a**5*x**5 - 3*a* 
*4*i*x**4 - 2*a**3*x**3 - 2*a**2*i*x**2 - 3*a*x + i),x)*a + 2*int(( - sqrt 
(a**2*x**2 + 1)*x**4)/(a**5*x**5 - 3*a**4*i*x**4 - 2*a**3*x**3 - 2*a**2*i* 
x**2 - 3*a*x + i),x)*a**5 - 6*int(( - sqrt(a**2*x**2 + 1)*x**3)/(a**5*x**5 
 - 3*a**4*i*x**4 - 2*a**3*x**3 - 2*a**2*i*x**2 - 3*a*x + i),x)*a**4*i - 22 
*int(( - sqrt(a**2*x**2 + 1)*x)/(a**5*x**5 - 3*a**4*i*x**4 - 2*a**3*x**3 - 
 2*a**2*i*x**2 - 3*a*x + i),x)*a**2*i - 6*int((sqrt(a**2*x**2 + 1)*x)/(a** 
3*i*x**3 + 3*a**2*x**2 - 3*a*i*x - 1),x)*a**2 + 3*log(sqrt(a**2*x**2 + 1) 
- 1) - 3*log(sqrt(a**2*x**2 + 1) + 1))/6