3.309 \(\int \frac{e^{-4 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac{2 i \sqrt{1-i a x}}{a \sqrt{1+i a x}}+\frac{\sinh ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0376806, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5073, 47, 41, 215} \[ \frac{2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac{2 i \sqrt{1-i a x}}{a \sqrt{1+i a x}}+\frac{\sinh ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^((4*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]

[Out]

(((2*I)/3)*(1 - I*a*x)^(3/2))/(a*(1 + I*a*x)^(3/2)) - ((2*I)*Sqrt[1 - I*a*x])/(a*Sqrt[1 + I*a*x]) + ArcSinh[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 47

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0715867, size = 82, normalized size = 1.12 \[ \frac{2 i \left (\frac{2 \sqrt{1+i a x} \left (2 a^2 x^2+i a x+1\right )}{\sqrt{1-i a x} (a x-i)^2}+3 \sin ^{-1}\left (\frac{\sqrt{1-i a x}}{\sqrt{2}}\right )\right )}{3 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^((4*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]

[Out]

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

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Maple [B]  time = 0.069, size = 262, normalized size = 3.6 \begin{align*}{\frac{{\frac{i}{3}}}{{a}^{5}} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-4}}+{\frac{1}{3\,{a}^{4}} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-3}}+{\frac{{\frac{2\,i}{3}}}{{a}^{3}} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-2}}-{\frac{{\frac{2\,i}{3}}}{a} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{3}{2}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }x+{\ln \left ({ \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x)

[Out]

1/3*I/a^5/(x-I/a)^4*(a^2*(x-I/a)^2+2*I*a*(x-I/a))^(5/2)+1/3/a^4/(x-I/a)^3*(a^2*(x-I/a)^2+2*I*a*(x-I/a))^(5/2)+
2/3*I/a^3/(x-I/a)^2*(a^2*(x-I/a)^2+2*I*a*(x-I/a))^(5/2)-2/3*I/a*(a^2*(x-I/a)^2+2*I*a*(x-I/a))^(3/2)+(a^2*(x-I/
a)^2+2*I*a*(x-I/a))^(1/2)*x+ln((I*a+a^2*(x-I/a))/(a^2)^(1/2)+(a^2*(x-I/a)^2+2*I*a*(x-I/a))^(1/2))/(a^2)^(1/2)

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Maxima [B]  time = 1.54856, size = 146, normalized size = 2. \begin{align*} \frac{i \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{-3 i \, a^{4} x^{3} - 9 \, a^{3} x^{2} + 9 i \, a^{2} x + 3 \, a} + \frac{\operatorname{arsinh}\left (a x\right )}{a} - \frac{2 i \, \sqrt{a^{2} x^{2} + 1}}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} - \frac{7 i \, \sqrt{a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

I*(a^2*x^2 + 1)^(3/2)/(-3*I*a^4*x^3 - 9*a^3*x^2 + 9*I*a^2*x + 3*a) + arcsinh(a*x)/a - 2*I*sqrt(a^2*x^2 + 1)/(3
*a^3*x^2 - 6*I*a^2*x - 3*a) - 7*I*sqrt(a^2*x^2 + 1)/(3*I*a^2*x + 3*a)

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Fricas [A]  time = 1.9496, size = 204, normalized size = 2.79 \begin{align*} -\frac{8 \, a^{2} x^{2} - 16 i \, a x +{\left (3 \, a^{2} x^{2} - 6 i \, a x - 3\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right ) + \sqrt{a^{2} x^{2} + 1}{\left (8 \, a x - 4 i\right )} - 8}{3 \, a^{3} x^{2} - 6 i \, a^{2} x - 3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

-(8*a^2*x^2 - 16*I*a*x + (3*a^2*x^2 - 6*I*a*x - 3)*log(-a*x + sqrt(a^2*x^2 + 1)) + sqrt(a^2*x^2 + 1)*(8*a*x -
4*I) - 8)/(3*a^3*x^2 - 6*I*a^2*x - 3*a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{\left (i a x + 1\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*a*x)**4*(a**2*x**2+1)**(3/2),x)

[Out]

Integral((a**2*x**2 + 1)**(3/2)/(I*a*x + 1)**4, x)

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Giac [A]  time = 1.15249, size = 32, normalized size = 0.44 \begin{align*} -\frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

-log(-x*abs(a) + sqrt(a^2*x^2 + 1))/abs(a)