\(\int e^{n \arctan (a x)} x^m (c+a^2 c x^2) \, dx\) [378]

Optimal result

Integrand size = 22, antiderivative size = 49 \[ \int e^{n \arctan (a x)} x^m \left (c+a^2 c x^2\right ) \, dx=\frac {c x^{1+m} \operatorname {AppellF1}\left (1+m,-1-\frac {i n}{2},-1+\frac {i n}{2},2+m,i a x,-i a x\right )}{1+m} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5605, 150}

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.

\(\displaystyle \int x^m \left (a^2 c x^2+c\right ) e^{n \arctan (a x)} \, dx\)

\(\Big \downarrow \) 5605

\(\displaystyle c \int x^m (1-i a x)^{\frac {i n}{2}+1} (i a x+1)^{1-\frac {i n}{2}}dx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {c x^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {i n}{2}-1,\frac {i n}{2}-1,m+2,i a x,-i a x\right )}{m+1}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> 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] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 5605

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[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] && (Integer 
Q[p] || GtQ[c, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c*(int(x**m*e**(atan(a*x)*n)*x**2,x)*a**2 + int(x**m*e**(atan(a*x)*n),x))