3.4.9 \(\int \frac {e^{-4 i \text {ArcTan}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\) [309]

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]

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

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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {5181, 49, 41, 221} \begin {gather*} \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 {gather*}

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 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 49

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 221

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]

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^{-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*}

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Mathematica [A]
time = 0.05, size = 82, normalized size = 1.12 \begin {gather*} \frac {2 i \left (\frac {2 \sqrt {1+i a x} \left (1+i a x+2 a^2 x^2\right )}{\sqrt {1-i a x} (-i+a x)^2}+3 \text {ArcSin}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{3 a} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (57 ) = 114\).
time = 0.09, size = 305, normalized size = 4.18

method	result	size
default	\(\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{3 a \left (x -\frac {i}{a}\right )^{4}}-\frac {i a \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{3}}{a^{4}}\)	\(305\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (51) = 102\).
time = 0.47, size = 107, normalized size = 1.47 \begin {gather*} \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 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} - \frac {7 i \, \sqrt {a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \end {gather*}

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 - 2/3*I*sqrt(a^2*x^2 + 1)/
(a^3*x^2 - 2*I*a^2*x - a) - 7*I*sqrt(a^2*x^2 + 1)/(3*I*a^2*x + 3*a)

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Fricas [A]
time = 5.12, size = 86, normalized size = 1.18 \begin {gather*} -\frac {8 \, a^{2} x^{2} - 16 i \, a x + 3 \, {\left (a^{2} x^{2} - 2 i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 4 \, \sqrt {a^{2} x^{2} + 1} {\left (2 \, a x - i\right )} - 8}{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/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a x - i\right )^{4}}\, dx \end {gather*}

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)/(a*x - I)**4, x)

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Giac [A]
time = 0.46, size = 24, normalized size = 0.33 \begin {gather*} -\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \end {gather*}

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)

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Mupad [B]
time = 0.10, size = 93, normalized size = 1.27 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {8\,\sqrt {a^2\,x^2+1}}{3\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{3\,\left (-a^4\,x^2+a^3\,x\,2{}\mathrm {i}+a^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x*(a^2)^(1/2))/(a^2)^(1/2) + (8*(a^2*x^2 + 1)^(1/2))/(3*(((a^2)^(1/2)*1i)/a - x*(a^2)^(1/2))*(a^2)^(1/2)
) + (a*(a^2*x^2 + 1)^(1/2)*4i)/(3*(a^3*x*2i + a^2 - a^4*x^2))

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