\(\int \frac {e^{n \arctan (a+b x)}}{x^2} \, dx\) [242]

Optimal result

Integrand size = 14, antiderivative size = 128 \[ \int \frac {e^{n \arctan (a+b x)}}{x^2} \, dx=-\frac {4 b (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i n}{2},2+\frac {i n}{2},\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i+a)^2 (2 i-n)} \]

[Out]

-4*b*(1-I*a-I*b*x)^(1+1/2*I*n)*(1+I*a+I*b*x)^(-1-1/2*I*n)*hypergeom([2, 1+1/2*I*n],[2+1/2*I*n],(I-a)*(1-I*a-I*
b*x)/(I+a)/(1+I*a+I*b*x))/(I+a)^2/(2*I-n)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5203, 133} \[ \int \frac {e^{n \arctan (a+b x)}}{x^2} \, dx=-\frac {4 b (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{-1-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,\frac {i n}{2}+1,\frac {i n}{2}+2,\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(a+i)^2 (-n+2 i)} \]

[In]

Int[E^(n*ArcTan[a + b*x])/x^2,x]

[Out]

(-4*b*(1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(-1 - (I/2)*n)*Hypergeometric2F1[2, 1 + (I/2)*n, 2 + (
I/2)*n, ((I - a)*(1 - I*a - I*b*x))/((I + a)*(1 + I*a + I*b*x))])/((I + a)^2*(2*I - n))

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 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 \frac {(1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x^2} \, dx \\ & = -\frac {4 b (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i n}{2},2+\frac {i n}{2},\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i+a)^2 (2 i-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int \frac {e^{n \arctan (a+b x)}}{x^2} \, dx=-\frac {4 i b (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i n}{2},2+\frac {i n}{2},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )}{(i+a)^2 (-2 i+n) (-i+a+b x)} \]

[In]

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

[Out]

((-4*I)*b*((-I)*(I + a + b*x))^(1 + (I/2)*n)*Hypergeometric2F1[2, 1 + (I/2)*n, 2 + (I/2)*n, (1 + a^2 - I*b*x +
a*b*x)/(1 + a^2 + I*b*x + a*b*x)])/((I + a)^2*(-2*I + n)*(1 + I*a + I*b*x)^((I/2)*n)*(-I + a + b*x))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(exp(n*atan(a + b*x))/x**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a+b x)}}{x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(exp(n*atan(a + b*x))/x^2,x)

[Out]

int(exp(n*atan(a + b*x))/x^2, x)