\(\int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx\) [317]

Optimal result

Integrand size = 25, antiderivative size = 86 \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{a (i-a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 45} \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{a (-a x+i) \sqrt {a^2 c x^2+c}} \]

[In]

Int[1/(E^((3*I)*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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])

Rule 5184

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{-3 i \arctan (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-i a x}{(1+i a x)^2} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \left (-\frac {2}{(-i+a x)^2}+\frac {i}{-i+a x}\right ) \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{a (i-a x) \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (-\frac {2}{a (i-a x)}+\frac {i \log (i-a x)}{a}\right )}{\sqrt {c+a^2 c x^2}} \]

[In]

Integrate[1/(E^((3*I)*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {\left (-i \ln \left (-a x +i\right ) a x -\ln \left (-a x +i\right )-2\right ) \sqrt {c \left (a^{2} x^{2}+1\right )}}{\sqrt {a^{2} x^{2}+1}\, c \left (-a x +i\right ) a}\)	\(66\)
risch	\(\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a \left (a x -i\right )}+\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x -i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\)	\(76\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (71) = 142\).

Time = 0.29 (sec) , antiderivative size = 357, normalized size of antiderivative = 4.15 \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {{\left (-i \, a^{3} c x^{3} - a^{2} c x^{2} - i \, a c x - c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) + {\left (i \, a^{3} c x^{3} + a^{2} c x^{2} + i \, a c x + c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) - 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (a^{3} c x^{3} - i \, a^{2} c x^{2} + a c x - i \, c\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {i \, \log \left (i \, a x + 1\right )}{a \sqrt {c}} + \frac {2}{a^{2} \sqrt {c} x - i \, a \sqrt {c}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c} {\left (i \, a x + 1\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\left (a^2\,x^2+1\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

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

[Out]

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