\(\int e^{n \arctan (a+b x)} x \, dx\) [239]

Optimal result

Integrand size = 12, antiderivative size = 147 \[ \int e^{n \arctan (a+b x)} x \, dx=\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 b^2}+\frac {2^{-\frac {i n}{2}} (2 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{b^2 (2 i-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5203, 81, 71} \[ \int e^{n \arctan (a+b x)} x \, dx=\frac {2^{-\frac {i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{b^2 (-n+2 i)}+\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 b^2} \]

[In]

Int[E^(n*ArcTan[a + b*x])*x,x]

[Out]

((1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/(2*b^2) + ((2*a + n)*(1 - I*a - I*b*x)^(1 +
(I/2)*n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (1 - I*a - I*b*x)/2])/(2^((I/2)*n)*b^2*(2*I - n)
)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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 (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx \\ & = \frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 b^2}-\frac {(2 a+n) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{2 b} \\ & = \frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 b^2}+\frac {2^{-\frac {i n}{2}} (2 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{b^2 (2 i-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int e^{n \arctan (a+b x)} x \, dx=\frac {i (-i (i+a+b x))^{1+\frac {i n}{2}} \left ((1+i a+i b x)^{-\frac {i n}{2}} (-i+a+b x)+\frac {2^{1-\frac {i n}{2}} (2 a+n) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )}{-2-i n}\right )}{2 b^2} \]

[In]

Integrate[E^(n*ArcTan[a + b*x])*x,x]

[Out]

((I/2)*((-I)*(I + a + b*x))^(1 + (I/2)*n)*((-I + a + b*x)/(1 + I*a + I*b*x)^((I/2)*n) + (2^(1 - (I/2)*n)*(2*a
+ n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)])/(-2 - I*n)))/b^2

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*e^(n*arctan(b*x + a)), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*exp(n*atan(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*e^(n*arctan(b*x + a)), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(x*exp(n*atan(a + b*x)),x)

[Out]

int(x*exp(n*atan(a + b*x)), x)