3.301 \(\int \frac{e^{5 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=50 \[ \frac{4 i}{a (1-i a x)}-\frac{2 i}{a (1-i a x)^2}+\frac{i \log (a x+i)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0427698, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5073, 43} \[ \frac{4 i}{a (1-i a x)}-\frac{2 i}{a (1-i a x)^2}+\frac{i \log (a x+i)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^((5*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(-2*I)/(a*(1 - I*a*x)^2) + (4*I)/(a*(1 - I*a*x)) + (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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{5 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx &=\int \frac{(1+i a x)^2}{(1-i a x)^3} \, dx\\ &=\int \left (\frac{4}{(1-i a x)^3}-\frac{4}{(1-i a x)^2}+\frac{1}{1-i a x}\right ) \, dx\\ &=-\frac{2 i}{a (1-i a x)^2}+\frac{4 i}{a (1-i a x)}+\frac{i \log (i+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0222498, size = 42, normalized size = 0.84 \[ \frac{i \left (4 i a x+(a x+i)^2 \log (a x+i)-2\right )}{a (a x+i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^((5*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(I*(-2 + (4*I)*a*x + (I + a*x)^2*Log[I + a*x]))/(a*(I + a*x)^2)

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Maple [A]  time = 0.049, size = 45, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( ax+i \right ) ^{2}} \left ( -4\,x-{\frac{2\,i}{a}} \right ) }+{\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)^5/(a^2*x^2+1)^3,x)

[Out]

(-4*x-2*I/a)/(a*x+I)^2+1/2*I/a*ln(a^2*x^2+1)+arctan(a*x)/a

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Maxima [A]  time = 1.48925, size = 85, normalized size = 1.7 \begin{align*} -\frac{32 \, a^{3} x^{3} - 48 i \, a^{2} x^{2} - 16 i}{8 \,{\left (a^{5} x^{4} + 2 \, a^{3} x^{2} + a\right )}} + \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)^5/(a^2*x^2+1)^3,x, algorithm="maxima")

[Out]

-1/8*(32*a^3*x^3 - 48*I*a^2*x^2 - 16*I)/(a^5*x^4 + 2*a^3*x^2 + a) + arctan(a*x)/a + 1/2*I*log(a^2*x^2 + 1)/a

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Fricas [A]  time = 1.85058, size = 115, normalized size = 2.3 \begin{align*} -\frac{4 \, a x -{\left (i \, a^{2} x^{2} - 2 \, a x - i\right )} \log \left (\frac{a x + i}{a}\right ) + 2 i}{a^{3} x^{2} + 2 i \, a^{2} x - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.589308, size = 41, normalized size = 0.82 \begin{align*} - \frac{4 a^{4} x + 2 i a^{3}}{a^{6} x^{2} + 2 i a^{5} x - a^{4}} + \frac{i \log{\left (a x + i \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**5/(a**2*x**2+1)**3,x)

[Out]

-(4*a**4*x + 2*I*a**3)/(a**6*x**2 + 2*I*a**5*x - a**4) + I*log(a*x + I)/a

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Giac [A]  time = 1.11795, size = 41, normalized size = 0.82 \begin{align*} \frac{i \log \left (a x + i\right )}{a} - \frac{2 \,{\left (2 \, a x + i\right )}}{{\left (a x + i\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^3,x, algorithm="giac")

[Out]

i*log(a*x + i)/a - 2*(2*a*x + i)/((a*x + i)^2*a)