3.334 \(\int \frac{e^{-2 i \tan ^{-1}(a x)}}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{x}{3 c \sqrt{a^2 c x^2+c}}+\frac{2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0548632, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5074, 653, 191} \[ \frac{x}{3 c \sqrt{a^2 c x^2+c}}+\frac{2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5074

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

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0333535, size = 78, normalized size = 1.44 \[ \frac{\sqrt{1-i a x} (2+i a x) \sqrt{a^2 x^2+1}}{3 a c \sqrt{1+i a x} (a x-i) \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.07, size = 137, normalized size = 2.5 \begin{align*} -{\frac{x}{c}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}-{\frac{2\,i}{a} \left ({\frac{{\frac{i}{3}}}{ac} \left ( x-{\frac{i}{a}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) }}}}+{\frac{{\frac{i}{3}}}{a{c}^{2}} \left ( 2\, \left ( x-{\frac{i}{a}} \right ){a}^{2}c+2\,iac \right ){\frac{1}{\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.38089, size = 101, normalized size = 1.87 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - 2 i\right )}}{3 \, a^{3} c^{2} x^{2} - 6 i \, a^{2} c^{2} x - 3 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*c*x^2 + c)*(a*x - 2*I)/(3*a^3*c^2*x^2 - 6*I*a^2*c^2*x - 3*a*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{2} x^{2} + 1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (i a x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22282, size = 105, normalized size = 1.94 \begin{align*} \frac{2 \, \sqrt{a^{2} c}{\left (\sqrt{c} i - 3 \, \sqrt{a^{2} c} x + 3 \, \sqrt{a^{2} c x^{2} + c}\right )} i^{2}}{3 \,{\left (\sqrt{c} i - \sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c}\right )}^{3} a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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