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

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 = 0.0549454, 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 = {5075, 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[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 5075

Int[E^(ArcTan[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/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]) && ILtQ[(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 = 0.0267966, size = 78, normalized size = 1.44 \[ \frac{(2-i a x) \sqrt{1+i a x} \sqrt{a^2 x^2+1}}{3 a c \sqrt{1-i a x} (a x+i) \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2 - I*a*x)*Sqrt[1 + 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 = 0.074, size = 398, normalized size = 7.4 \begin{align*} -{\frac{x}{c}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}+{\frac{1}{a} \left ( i\sqrt{-{a}^{2}}+a \right ) \left ( -{\frac{1}{3\,c}{\frac{1}{\sqrt{-{a}^{2}}}} \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c+2\,c\sqrt{-{a}^{2}} \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}}-{\frac{1}{3\,{c}^{2}} \left ( 2\, \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ){a}^{2}c+2\,c\sqrt{-{a}^{2}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c+2\,c\sqrt{-{a}^{2}} \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}}+{\frac{1}{a} \left ( i\sqrt{-{a}^{2}}-a \right ) \left ({\frac{1}{3\,c}{\frac{1}{\sqrt{-{a}^{2}}}} \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c-2\,c\sqrt{-{a}^{2}} \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}}+{\frac{1}{3\,{c}^{2}} \left ( 2\, \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ){a}^{2}c-2\,c\sqrt{-{a}^{2}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c-2\,c\sqrt{-{a}^{2}} \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.37992, 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+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{\left (i a x + 1\right )^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a^{2} x^{2} + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14482, size = 103, normalized size = 1.91 \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 )}}{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+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))/((sqrt(c)*i + sqrt(a^2*c)*x - sqrt(a^2*
c*x^2 + c))^3*a^2*c)