∫ e−iArcTan(ax) ________________________

Optimal. Leaf size=43 \[ -\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}} \]

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Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 31} \begin {gather*} -\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}} \end {gather*}

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

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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,

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]

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

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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}} \end {gather*}

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

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

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method	result	size

risch	\(-\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (-a x +i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\)	\(39\)
default	\(-\frac {i \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}\, c a}\)	\(42\)

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

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.35 \begin {gather*} -\frac {i \, \log \left (i \, a x + 1\right )}{a \sqrt {c}} \end {gather*}

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (35) = 70\).
time = 2.80, size = 253, normalized size = 5.88 \begin {gather*} \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) - \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) \end {gather*}

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

1/2*I*sqrt(1/(a^2*c))*log(1/8*((-I*a^6*x^2 - 2*a^5*x + 2*I*a^4)*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1) + (I*a^9

*c*x^4 + 2*a^8*c*x^3 + I*a^7*c*x^2 + 2*a^6*c*x)*sqrt(1/(a^2*c)))/(a^3*x^3 - I*a^2*x^2 + a*x - I)) - 1/2*I*sqrt

(1/(a^2*c))*log(1/8*((-I*a^6*x^2 - 2*a^5*x + 2*I*a^4)*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1) + (-I*a^9*c*x^4 -

2*a^8*c*x^3 - I*a^7*c*x^2 - 2*a^6*c*x)*sqrt(1/(a^2*c)))/(a^3*x^3 - I*a^2*x^2 + a*x - I))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x \sqrt {a^{2} c x^{2} + c} - i \sqrt {a^{2} c x^{2} + c}}\, dx \end {gather*}

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

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a^2\,x^2+1}}{\sqrt {c\,a^2\,x^2+c}\,\left (1+a\,x\,1{}\mathrm {i}\right )} \,d x \end {gather*}

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

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

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3.4.15 \(\int \frac {e^{-i \text {ArcTan}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\) [315]