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

Optimal. Leaf size=88 \[ \frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1}}{2 a c (a x+i) \sqrt{a^2 c x^2+c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0813627, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {5076, 5073, 44, 203} \[ \frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1}}{2 a c (a x+i) \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5076

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

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0307032, size = 51, normalized size = 0.58 \[ \frac{\sqrt{a^2 x^2+1} \left (\tan ^{-1}(a x)+\frac{1}{a x+i}\right )}{2 a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.138, size = 58, normalized size = 0.7 \begin{align*} -{\frac{-\arctan \left ( ax \right ){x}^{2}{a}^{2}-ax+i-\arctan \left ( ax \right ) }{2\,a{c}^{2}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ({a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 2.44164, size = 698, normalized size = 7.93 \begin{align*} \frac{{\left (-i \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - i \, a c^{2} x + c^{2}\right )} \sqrt{\frac{1}{a^{2} c^{3}}} \log \left (\frac{8 \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} a^{6} x +{\left (4 i \, a^{10} c^{2} x^{4} - 4 i \, a^{6} c^{2}\right )} \sqrt{\frac{1}{a^{2} c^{3}}}}{2 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) +{\left (i \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} + i \, a c^{2} x - c^{2}\right )} \sqrt{\frac{1}{a^{2} c^{3}}} \log \left (\frac{8 \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} a^{6} x +{\left (-4 i \, a^{10} c^{2} x^{4} + 4 i \, a^{6} c^{2}\right )} \sqrt{\frac{1}{a^{2} c^{3}}}}{2 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) + 4 i \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} x}{2 \,{\left (4 \, a^{3} c^{2} x^{3} + 4 i \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 4 i \, c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((-I*a^3*c^2*x^3 + a^2*c^2*x^2 - I*a*c^2*x + c^2)*sqrt(1/(a^2*c^3))*log(1/2*(8*sqrt(a^2*c*x^2 + c)*sqrt(a^
2*x^2 + 1)*a^6*x + (4*I*a^10*c^2*x^4 - 4*I*a^6*c^2)*sqrt(1/(a^2*c^3)))/(a^4*x^4 + 2*a^2*x^2 + 1)) + (I*a^3*c^2
*x^3 - a^2*c^2*x^2 + I*a*c^2*x - c^2)*sqrt(1/(a^2*c^3))*log(1/2*(8*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^6*x
+ (-4*I*a^10*c^2*x^4 + 4*I*a^6*c^2)*sqrt(1/(a^2*c^3)))/(a^4*x^4 + 2*a^2*x^2 + 1)) + 4*I*sqrt(a^2*c*x^2 + c)*s
qrt(a^2*x^2 + 1)*x)/(4*a^3*c^2*x^3 + 4*I*a^2*c^2*x^2 + 4*a*c^2*x + 4*I*c^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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