Integrand size = 35, antiderivative size = 60 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {i e^{i n \arctan (a x)} (1-i a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \] Output:
I*exp(I*n*arctan(a*x))*(1-I*a*n*x)/a^3/c/n/(-n^2+1)/((a^2*c*x^2+c)^(1/2*n^ 2))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {e^{i n \arctan (a x)} (i+a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (-1+n^2\right )} \] Input:
Integrate[E^(I*n*ArcTan[a*x])*x^2*(c + a^2*c*x^2)^(-1 - n^2/2),x]
Output:
-((E^(I*n*ArcTan[a*x])*(I + a*n*x))/(a^3*c*n*(-1 + n^2)*(c + a^2*c*x^2)^(n ^2/2)))
Time = 0.48 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {5602}
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.
\(\displaystyle \int x^2 e^{i n \arctan (a x)} \left (a^2 c x^2+c\right )^{-\frac {n^2}{2}-1} \, dx\) |
\(\Big \downarrow \) 5602 |
\(\displaystyle \frac {i (1-i a n x) e^{i n \arctan (a x)} \left (a^2 c x^2+c\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )}\) |
Input:
Int[E^(I*n*ArcTan[a*x])*x^2*(c + a^2*c*x^2)^(-1 - n^2/2),x]
Output:
(I*E^(I*n*ArcTan[a*x])*(1 - I*a*n*x))/(a^3*c*n*(1 - n^2)*(c + a^2*c*x^2)^( n^2/2))
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^2*((c_) + (d_.)*(x_)^2)^(p_.), x_Symb ol] :> Simp[(-(1 - a*n*x))*(c + d*x^2)^(p + 1)*(E^(n*ArcTan[a*x])/(a*d*n*(n ^2 + 1))), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && !IntegerQ[I*n]
Time = 5.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98
method | result | size |
orering | \(-\frac {\left (n a x +i\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\) | \(59\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (n a x +i\right ) {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\) | \(62\) |
parallelrisch | \(-\frac {{\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} x^{3} a^{3} n +i {\mathrm e}^{i n \arctan \left (a x \right )} x^{2} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} a^{2}+{\mathrm e}^{i n \arctan \left (a x \right )} x \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} a n +i {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\) | \(149\) |
Input:
int(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x,method=_RETURNVE RBOSE)
Output:
-(n*a*x+I)/a^3/n/(n^2-1)*(a^2*x^2+1)*exp(I*n*arctan(a*x))*(a^2*c*x^2+c)^(- 1-1/2*n^2)
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.30 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {{\left (a^{3} n x^{3} + i \, a^{2} x^{2} + a n x + i\right )} {\left (a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1}}{{\left (a^{3} n^{3} - a^{3} n\right )} \left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}} \] Input:
integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm ="fricas")
Output:
-(a^3*n*x^3 + I*a^2*x^2 + a*n*x + I)*(a^2*c*x^2 + c)^(-1/2*n^2 - 1)/((a^3* n^3 - a^3*n)*(-(a*x + I)/(a*x - I))^(1/2*n))
Timed out. \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\text {Timed out} \] Input:
integrate(exp(I*n*atan(a*x))*x**2*(a**2*c*x**2+c)**(-1-1/2*n**2),x)
Output:
Timed out
\[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1} x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm ="maxima")
Output:
integrate((a^2*c*x^2 + c)^(-1/2*n^2 - 1)*x^2*e^(I*n*arctan(a*x)), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (51) = 102\).
Time = 0.43 (sec) , antiderivative size = 333, normalized size of antiderivative = 5.55 \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {a^{3} n x^{3} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + i\right ) - \frac {1}{2} \, n^{2} \log \left (a x - i\right ) - \frac {1}{2} \, n^{2} \log \left (c\right ) + \frac {1}{2} i \, \pi n - \frac {1}{2} \, n \log \left (a x + i\right ) + \frac {1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + i \, a^{2} x^{2} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + i\right ) - \frac {1}{2} \, n^{2} \log \left (a x - i\right ) - \frac {1}{2} \, n^{2} \log \left (c\right ) + \frac {1}{2} i \, \pi n - \frac {1}{2} \, n \log \left (a x + i\right ) + \frac {1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + a n x e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + i\right ) - \frac {1}{2} \, n^{2} \log \left (a x - i\right ) - \frac {1}{2} \, n^{2} \log \left (c\right ) + \frac {1}{2} i \, \pi n - \frac {1}{2} \, n \log \left (a x + i\right ) + \frac {1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + i \, e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + i\right ) - \frac {1}{2} \, n^{2} \log \left (a x - i\right ) - \frac {1}{2} \, n^{2} \log \left (c\right ) + \frac {1}{2} i \, \pi n - \frac {1}{2} \, n \log \left (a x + i\right ) + \frac {1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )}}{a^{3} n^{3} - a^{3} n} \] Input:
integrate(exp(I*n*arctan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm ="giac")
Output:
-(a^3*n*x^3*e^(-1/2*n^2*log(a*x + I) - 1/2*n^2*log(a*x - I) - 1/2*n^2*log( c) + 1/2*I*pi*n - 1/2*n*log(a*x + I) + 1/2*n*log(a*x - I) - log(a*x + I) - log(a*x - I) - log(c)) + I*a^2*x^2*e^(-1/2*n^2*log(a*x + I) - 1/2*n^2*log (a*x - I) - 1/2*n^2*log(c) + 1/2*I*pi*n - 1/2*n*log(a*x + I) + 1/2*n*log(a *x - I) - log(a*x + I) - log(a*x - I) - log(c)) + a*n*x*e^(-1/2*n^2*log(a* x + I) - 1/2*n^2*log(a*x - I) - 1/2*n^2*log(c) + 1/2*I*pi*n - 1/2*n*log(a* x + I) + 1/2*n*log(a*x - I) - log(a*x + I) - log(a*x - I) - log(c)) + I*e^ (-1/2*n^2*log(a*x + I) - 1/2*n^2*log(a*x - I) - 1/2*n^2*log(c) + 1/2*I*pi* n - 1/2*n*log(a*x + I) + 1/2*n*log(a*x - I) - log(a*x + I) - log(a*x - I) - log(c)))/(a^3*n^3 - a^3*n)
Timed out. \[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{{\left (c\,a^2\,x^2+c\right )}^{\frac {n^2}{2}+1}} \,d x \] Input:
int((x^2*exp(n*atan(a*x)*1i))/(c + a^2*c*x^2)^(n^2/2 + 1),x)
Output:
int((x^2*exp(n*atan(a*x)*1i))/(c + a^2*c*x^2)^(n^2/2 + 1), x)
\[ \int e^{i n \arctan (a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {\int \frac {e^{\mathit {atan} \left (a x \right ) i n} x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {n^{2}}{2}} a^{2} x^{2}+\left (a^{2} c \,x^{2}+c \right )^{\frac {n^{2}}{2}}}d x}{c} \] Input:
int(exp(I*n*atan(a*x))*x^2*(a^2*c*x^2+c)^(-1-1/2*n^2),x)
Output:
int((e**(atan(a*x)*i*n)*x**2)/((a**2*c*x**2 + c)**(n**2/2)*a**2*x**2 + (a* *2*c*x**2 + c)**(n**2/2)),x)/c