\(\int e^{n \arctan (a+b x)} x^m \, dx\) [236]

Optimal result

Integrand size = 14, antiderivative size = 140 \[ \int e^{n \arctan (a+b x)} x^m \, dx=\frac {x^{1+m} (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (1-\frac {b x}{i-a}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{i+a}\right )^{-\frac {i n}{2}} \operatorname {AppellF1}\left (1+m,-\frac {i n}{2},\frac {i n}{2},2+m,-\frac {b x}{i+a},\frac {b x}{i-a}\right )}{1+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5203, 140, 138} \[ \int e^{n \arctan (a+b x)} x^m \, dx=\frac {x^{m+1} (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \left (1-\frac {b x}{-a+i}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{a+i}\right )^{-\frac {i n}{2}} \operatorname {AppellF1}\left (m+1,-\frac {i n}{2},\frac {i n}{2},m+2,-\frac {b x}{a+i},\frac {b x}{i-a}\right )}{m+1} \]

[In]

Int[E^(n*ArcTan[a + b*x])*x^m,x]

[Out]

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

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 140

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

Rule 5203

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1 -
I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

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

Mathematica [F]

\[ \int e^{n \arctan (a+b x)} x^m \, dx=\int e^{n \arctan (a+b x)} x^m \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(x^m*exp(n*atan(a + b*x)),x)

[Out]

int(x^m*exp(n*atan(a + b*x)), x)