\(\int e^{-2 i p \arctan (a x)} (c+a^2 c x^2)^p \, dx\) [372]

Optimal result

Integrand size = 24, antiderivative size = 53 \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i (1-i a x)^{1+2 p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p}{a (1+2 p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5184, 5181, 32} \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {i (1-i a x)^{2 p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p}{a (2 p+1)} \]

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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}& = \left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int e^{-2 i p \arctan (a x)} \left (1+a^2 x^2\right )^p \, dx \\ & = \left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int (1-i a x)^{2 p} \, dx \\ & = \frac {i (1-i a x)^{1+2 p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p}{a (1+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {e^{-2 i p \arctan (a x)} (i+a x) \left (c+a^2 c x^2\right )^p}{a+2 a p} \]

[In]

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

[Out]

((I + a*x)*(c + a^2*c*x^2)^p)/(E^((2*I)*p*ArcTan[a*x])*(a + 2*a*p))

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\left (a x +i\right ) \left (a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 i p \arctan \left (a x \right )}}{a \left (1+2 p \right )}\)	\(41\)
parallelrisch	\(-\frac {\left (-x \left (a^{2} c \,x^{2}+c \right )^{p} a -i \left (a^{2} c \,x^{2}+c \right )^{p}\right ) {\mathrm e}^{-2 i p \arctan \left (a x \right )}}{a \left (1+2 p \right )}\)	\(56\)
risch	\(\frac {\left (\left (a x +i\right )^{p}\right )^{2} c^{p} \left (a x +i\right ) {\mathrm e}^{-\frac {i p \pi \left (\operatorname {csgn}\left (a x +i\right )^{3}-\operatorname {csgn}\left (a x +i\right )^{2} \operatorname {csgn}\left (i \left (a x +i\right )\right )+\operatorname {csgn}\left (i \left (a x +i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )-\operatorname {csgn}\left (i \left (a x +i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2}+\operatorname {csgn}\left (a x -i\right )^{3}+\operatorname {csgn}\left (a x -i\right )^{2} \operatorname {csgn}\left (i \left (a x -i\right )\right )-\operatorname {csgn}\left (i \left (a x -i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2}+\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )-\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{3}-\operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2}+\operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{3}-\operatorname {csgn}\left (a x +i\right )^{2}+\operatorname {csgn}\left (a x +i\right ) \operatorname {csgn}\left (i \left (a x +i\right )\right )-\operatorname {csgn}\left (a x -i\right )^{2}-\operatorname {csgn}\left (a x -i\right ) \operatorname {csgn}\left (i \left (a x -i\right )\right )+2\right )}{2}}}{\left (1+2 p \right ) a}\)	\(403\)

[In]

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

[Out]

(I+a*x)/a/(1+2*p)*(a^2*c*x^2+c)^p/exp(2*I*p*arctan(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {{\left (a x + i\right )} {\left (a^{2} c x^{2} + c\right )}^{p} \left (-\frac {a x + i}{a x - i}\right )^{p}}{2 \, a p + a} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\begin {cases} \frac {x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{i \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + a e^{2 i p \operatorname {atan}{\left (a x \right )}}} + \frac {i \left (a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 i p \operatorname {atan}{\left (a x \right )}} + a e^{2 i p \operatorname {atan}{\left (a x \right )}}} & \text {otherwise} \end {cases} \]

[In]

integrate((a**2*c*x**2+c)**p/exp(2*I*p*atan(a*x)),x)

[Out]

Piecewise((x/sqrt(c), Eq(a, 0) & Eq(p, -1/2)), (c**p*x, Eq(a, 0)), (Integral(exp(I*atan(a*x))/sqrt(c*(a**2*x**
2 + 1)), x), Eq(p, -1/2)), (a*x*(a**2*c*x**2 + c)**p/(2*a*p*exp(2*I*p*atan(a*x)) + a*exp(2*I*p*atan(a*x))) + I
*(a**2*c*x**2 + c)**p/(2*a*p*exp(2*I*p*atan(a*x)) + a*exp(2*I*p*atan(a*x))), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.43 \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {{\left (a c^{p} x + i \, c^{p}\right )} {\left (a^{2} x^{2} + 1\right )}^{p} \cos \left (2 \, p \arctan \left (a x\right )\right ) - {\left (i \, a c^{p} x - c^{p}\right )} {\left (a^{2} x^{2} + 1\right )}^{p} \sin \left (2 \, p \arctan \left (a x\right )\right )}{2 \, a p + a} \]

[In]

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

[Out]

((a*c^p*x + I*c^p)*(a^2*x^2 + 1)^p*cos(2*p*arctan(a*x)) - (I*a*c^p*x - c^p)*(a^2*x^2 + 1)^p*sin(2*p*arctan(a*x
)))/(2*a*p + a)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int e^{-2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\left (\frac {x\,{\mathrm {e}}^{-p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}}{2\,p+1}+\frac {{\mathrm {e}}^{-p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}\,1{}\mathrm {i}}{a\,\left (2\,p+1\right )}\right )\,{\left (c\,a^2\,x^2+c\right )}^p \]

[In]

int(exp(-p*atan(a*x)*2i)*(c + a^2*c*x^2)^p,x)

[Out]

((x*exp(-p*atan(a*x)*2i))/(2*p + 1) + (exp(-p*atan(a*x)*2i)*1i)/(a*(2*p + 1)))*(c + a^2*c*x^2)^p