3.4.22 \(\int \frac {e^{2 i \text {ArcTan}(a x)}}{(1+a^2 x^2)^{3/2}} \, dx\) [322]

Optimal. Leaf size=67 \[ -\frac {i \sqrt {1+i a x}}{3 a (1-i a x)^{3/2}}-\frac {i \sqrt {1+i a x}}{3 a \sqrt {1-i a x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5181, 47, 37} \begin {gather*} -\frac {i \sqrt {1+i a x}}{3 a \sqrt {1-i a x}}-\frac {i \sqrt {1+i a x}}{3 a (1-i a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/3*I)*Sqrt[1 + I*a*x])/(a*(1 - I*a*x)^(3/2)) - ((I/3)*Sqrt[1 + I*a*x])/(a*Sqrt[1 - I*a*x])

Rule 37

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

Rule 47

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

Rule 5181

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 48, normalized size = 0.72 \begin {gather*} \frac {(2-i a x) \sqrt {1+i a x}}{3 a \sqrt {1-i a x} (i+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2 - I*a*x)*Sqrt[1 + I*a*x])/(3*a*Sqrt[1 - I*a*x]*(I + a*x))

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (49 ) = 98\).
time = 0.08, size = 104, normalized size = 1.55

method	result	size
trager	\(\frac {a^{3} x^{3}+3 a x -2 i}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\)	\(31\)
meijerg	\(\frac {x \left (2 a^{2} x^{2}+3\right )}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {4 i \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3 a \sqrt {\pi }}-\frac {a^{2} x^{3}}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\)	\(76\)
default	\(\frac {x}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x}{3 \sqrt {a^{2} x^{2}+1}}-a^{2} \left (-\frac {x}{2 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {\frac {x}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x}{3 \sqrt {a^{2} x^{2}+1}}}{2 a^{2}}\right )-\frac {2 i}{3 a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^2/(a^2*x^2+1)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 45, normalized size = 0.67 \begin {gather*} \frac {x}{3 \, \sqrt {a^{2} x^{2} + 1}} + \frac {2 \, x}{3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2 i}{3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.56, size = 51, normalized size = 0.76 \begin {gather*} \frac {a^{2} x^{2} + 2 i \, a x + \sqrt {a^{2} x^{2} + 1} {\left (a x + 2 i\right )} - 1}{3 \, {\left (a^{3} x^{2} + 2 i \, a^{2} x - a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a^{2} x^{2}}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx - \int \left (- \frac {2 i a x}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx - \int \left (- \frac {1}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.44, size = 67, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} - 3 \, {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{2} + 3 \, a {\left (-i \, \sqrt {a^{2} + \frac {1}{x^{2}}} + \frac {i}{x}\right )}\right )}}{3 \, {\left (i \, a + \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.50, size = 33, normalized size = 0.49 \begin {gather*} \frac {\sqrt {a^2\,x^2+1}\,\left (-2+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,a\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x*1i + 1)^2/(a^2*x^2 + 1)^(5/2),x)

[Out]

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

________________________________________________________________________________________