\(\int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx\) [365]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,1-\frac {i n}{2},1+\frac {i n}{2},2+m,i a x,-i a x\right )}{c (1+m)} \]

[Out]

x^(1+m)*AppellF1(1+m,1+1/2*I*n,1-1/2*I*n,2+m,-I*a*x,I*a*x)/c/(1+m)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5190, 138} \[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,1-\frac {i n}{2},\frac {i n}{2}+1,m+2,i a x,-i a x\right )}{c (m+1)} \]

[In]

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

[Out]

(x^(1 + m)*AppellF1[1 + m, 1 - (I/2)*n, 1 + (I/2)*n, 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 5190

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\int x^m (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, dx}{c} \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,1-\frac {i n}{2},1+\frac {i n}{2},2+m,i a x,-i a x\right )}{c (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.88 \[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\frac {e^{n \arctan (a x)} \left (1-e^{2 i \arctan (a x)}\right )^{-m} \left (1+e^{2 i \arctan (a x)}\right )^m x^m \operatorname {AppellF1}\left (-\frac {i n}{2},m,-m,1-\frac {i n}{2},-e^{2 i \arctan (a x)},e^{2 i \arctan (a x)}\right )}{a c n} \]

[In]

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

[Out]

(E^(n*ArcTan[a*x])*(1 + E^((2*I)*ArcTan[a*x]))^m*x^m*AppellF1[(-1/2*I)*n, m, -m, 1 - (I/2)*n, -E^((2*I)*ArcTan
[a*x]), E^((2*I)*ArcTan[a*x])])/(a*c*(1 - E^((2*I)*ArcTan[a*x]))^m*n)

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{a^{2} c \,x^{2}+c}d x\]

[In]

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c),x)

[Out]

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c),x)

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

integral(x^m*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

integrate(exp(n*atan(a*x))*x**m/(a**2*c*x**2+c),x)

[Out]

Integral(x**m*exp(n*atan(a*x))/(a**2*x**2 + 1), x)/c

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

integrate(x^m*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)

Giac [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\int { \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{c+a^2 c x^2} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]

[In]

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2),x)

[Out]

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2), x)