3.305 \(\int \frac{e^{i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=15 \[ \frac{i \log (a x+i)}{a} \]

[Out]

(I*Log[I + a*x])/a

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Rubi [A]  time = 0.02944, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5073, 31} \[ \frac{i \log (a x+i)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Log[I + a*x])/a

Rule 5073

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - I*a*x)^(p + (I*n
)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c,
0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{e^{i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx &=\int \frac{1}{1-i a x} \, dx\\ &=\frac{i \log (i+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0052172, size = 15, normalized size = 1. \[ \frac{i \log (a x+i)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Log[I + a*x])/a

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Maple [A]  time = 0.033, size = 26, normalized size = 1.7 \begin{align*}{\frac{{\frac{i}{2}}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}+{\frac{\arctan \left ( ax \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.49732, size = 32, normalized size = 2.13 \begin{align*} \frac{\arctan \left (a x\right )}{a} + \frac{i \, \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*x)/a + 1/2*I*log(a^2*x^2 + 1)/a

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Fricas [A]  time = 1.78238, size = 30, normalized size = 2. \begin{align*} \frac{i \, \log \left (\frac{a x + i}{a}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log((a*x + I)/a)/a

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Sympy [A]  time = 0.089788, size = 12, normalized size = 0.8 \begin{align*} \frac{i \log{\left (a^{2} x + i a \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log(a**2*x + I*a)/a

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Giac [A]  time = 1.09913, size = 18, normalized size = 1.2 \begin{align*} \frac{i \log \left (-a i x + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

i*log(-a*i*x + 1)/a