Integrand size = 24, antiderivative size = 53 \[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=-\frac {i (1+i a x)^{1+2 p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p}{a (1+2 p)} \]
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {e^{2 i p \arctan (a x)} (-i+a x) \left (c+a^2 c x^2\right )^p}{a+2 a p} \]
Time = 0.31 (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 = {5599, 5596, 17}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{2 i p \arctan (a x)} \left (a^2 c x^2+c\right )^p \, dx\) |
\(\Big \downarrow \) 5599 |
\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int e^{2 i p \arctan (a x)} \left (a^2 x^2+1\right )^pdx\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \int (i a x+1)^{2 p}dx\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {i (1+i a x)^{2 p+1} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p}{a (2 p+1)}\) |
3.4.73.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[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])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S imp[c^IntPart[p]*((c + d*x^2)^FracPart[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] && E qQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
Time = 1.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(-\frac {\left (-a x +i\right ) {\mathrm e}^{2 i p \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}}{a \left (1+2 p \right )}\) | \(41\) |
parallelrisch | \(-\frac {-{\mathrm e}^{2 i p \arctan \left (a x \right )} x \left (a^{2} c \,x^{2}+c \right )^{p} a +i {\mathrm e}^{2 i p \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}}{a \left (1+2 p \right )}\) | \(63\) |
risch | \(\frac {\left (a x +i\right )^{p} c^{p} \left (a x -i\right )^{2 p} \left (a x +i\right )^{-p} \left (a x -i\right ) {\mathrm e}^{-\frac {i p \pi \left (-\operatorname {csgn}\left (a x +i\right )^{3}+\operatorname {csgn}\left (a x +i\right )^{2} \operatorname {csgn}\left (i \left (a x +i\right )\right )+\operatorname {csgn}\left (i \left (a x +i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )-\operatorname {csgn}\left (i \left (a x +i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2}-\operatorname {csgn}\left (a x -i\right )^{3}-\operatorname {csgn}\left (a x -i\right )^{2} \operatorname {csgn}\left (i \left (a x -i\right )\right )-\operatorname {csgn}\left (i \left (a x -i\right )\right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{2}+\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )-\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right )^{3}-\operatorname {csgn}\left (i \left (a x -i\right ) \left (a x +i\right )\right ) \operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{2}+\operatorname {csgn}\left (i c \left (a x +i\right ) \left (a x -i\right )\right )^{3}+\operatorname {csgn}\left (a x +i\right )^{2}-\operatorname {csgn}\left (a x +i\right ) \operatorname {csgn}\left (i \left (a x +i\right )\right )+\operatorname {csgn}\left (a x -i\right )^{2}+\operatorname {csgn}\left (a x -i\right ) \operatorname {csgn}\left (i \left (a x -i\right )\right )-2\right )}{2}}}{\left (1+2 p \right ) a}\) | \(411\) |
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\frac {{\left (a x - i\right )} {\left (a^{2} c x^{2} + c\right )}^{p}}{{\left (2 \, a p + a\right )} \left (-\frac {a x + i}{a x - i}\right )^{p}} \]
\[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\begin {cases} \frac {x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{- i \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (a^{2} c x^{2} + c\right )^{p} e^{2 i p \operatorname {atan}{\left (a x \right )}}}{2 a p + a} - \frac {i \left (a^{2} c x^{2} + c\right )^{p} e^{2 i p \operatorname {atan}{\left (a x \right )}}}{2 a p + a} & \text {otherwise} \end {cases} \]
Piecewise((x/sqrt(c), Eq(a, 0) & Eq(p, -1/2)), (c**p*x, Eq(a, 0)), (Integr al(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*exp(2*I*p*atan(a*x))/(2*a*p + a) - I*(a**2*c*x**2 + c)**p* exp(2*I*p*atan(a*x))/(2*a*p + a), True))
\[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (2 i \, p \arctan \left (a x\right )\right )} \,d x } \]
\[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (2 i \, p \arctan \left (a x\right )\right )} \,d x } \]
Time = 0.68 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int e^{2 i p \arctan (a x)} \left (c+a^2 c x^2\right )^p \, dx=\left (\frac {x\,{\mathrm {e}}^{p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}}{2\,p+1}-\frac {{\mathrm {e}}^{p\,\mathrm {atan}\left (a\,x\right )\,2{}\mathrm {i}}\,1{}\mathrm {i}}{a\,\left (2\,p+1\right )}\right )\,{\left (c\,a^2\,x^2+c\right )}^p \]