3.1.4 \(\int e^{i \text {ArcTan}(a x)} x \, dx\) [4]

Optimal. Leaf size=42 \[ \frac {(2+i a x) \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \sinh ^{-1}(a x)}{2 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5168, 794, 221} \begin {gather*} \frac {(2+i a x) \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \sinh ^{-1}(a x)}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(I*ArcTan[a*x])*x,x]

[Out]

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

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 5168

Int[E^(ArcTan[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n
- 1)/2)*Sqrt[1 + a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 38, normalized size = 0.90 \begin {gather*} \frac {(2+i a x) \sqrt {1+a^2 x^2}-i \sinh ^{-1}(a x)}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(I*ArcTan[a*x])*x,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (34 ) = 68\).
time = 0.06, size = 72, normalized size = 1.71

method	result	size
risch	\(\frac {i \left (a x -2 i\right ) \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {a^{2}}}\)	\(59\)
default	\(i a \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {\sqrt {a^{2} x^{2}+1}}{a^{2}}\)	\(72\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{2 a^{2} \sqrt {\pi }}+\frac {i \left (\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}} \sqrt {a^{2} x^{2}+1}}{a^{2}}-\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \arcsinh \left (a x \right )}{a^{3}}\right )}{2 a \sqrt {\pi }\, \sqrt {a^{2}}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)/(a^2*x^2+1)^(1/2)*x,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 42, normalized size = 1.00 \begin {gather*} \frac {i \, \sqrt {a^{2} x^{2} + 1} x}{2 \, a} - \frac {i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} + \frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.34, size = 43, normalized size = 1.02 \begin {gather*} \frac {\sqrt {a^{2} x^{2} + 1} {\left (i \, a x + 2\right )} + i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 2.06, size = 51, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\\frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases} + \frac {i x \sqrt {a^{2} x^{2} + 1}}{2 a} - \frac {i \operatorname {asinh}{\left (a x \right )}}{2 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2/2, Eq(a**2, 0)), (sqrt(a**2*x**2 + 1)/a**2, True)) + I*x*sqrt(a**2*x**2 + 1)/(2*a) - I*asinh(a
*x)/(2*a**2)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 53, normalized size = 1.26 \begin {gather*} -\frac {1}{2} \, \sqrt {a^{2} x^{2} + 1} {\left (-\frac {i \, x}{a} - \frac {2}{a^{2}}\right )} + \frac {i \, \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 51, normalized size = 1.21 \begin {gather*} \frac {\left (\frac {1}{\sqrt {a^2}}+\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )\,\sqrt {a^2\,x^2+1}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{2\,a}}{\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________