\(\int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx\) [243]

Optimal result

Integrand size = 14, antiderivative size = 207 \[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 \left (1+a^2\right ) x^2}-\frac {2 b^2 (2 a-n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i n}{2},2+\frac {i n}{2},\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i-a) (i+a)^3 (2 i-n)} \]

[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)/(a^2+1)/x^2-2*b^2*(2*a-n)*(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)/
(I+a)^3/(2*I-n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5203, 98, 133} \[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=-\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 \left (a^2+1\right ) x^2}-\frac {2 b^2 (2 a-n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{-1-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (2,\frac {i n}{2}+1,\frac {i n}{2}+2,\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(-a+i) (a+i)^3 (-n+2 i)} \]

[In]

Int[E^(n*ArcTan[a + b*x])/x^3,x]

[Out]

-1/2*((1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/((1 + a^2)*x^2) - (2*b^2*(2*a - n)*(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)*(I + a)^3*(2*I - n))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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^3} \, 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 \left (1+a^2\right ) x^2}-\frac {(b (2 a-n)) \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}{2 \left (1+a^2\right )} \\ & = -\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 \left (1+a^2\right ) x^2}+\frac {2 b^2 (2 a-n) (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 \left (1+a^2\right ) (2 i-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84 \[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=-\frac {i (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{1+\frac {i n}{2}} \left ((i+a)^2 (-2 i+n) (-i+a+b x)^2+4 b^2 (-2 a+n) x^2 \operatorname {Hypergeometric2F1}\left (2,1+\frac {i n}{2},2+\frac {i n}{2},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{2 (-i+a) (i+a)^3 (-2 i+n) x^2 (-i+a+b x)} \]

[In]

Integrate[E^(n*ArcTan[a + b*x])/x^3,x]

[Out]

((-1/2*I)*((-I)*(I + a + b*x))^(1 + (I/2)*n)*((I + a)^2*(-2*I + n)*(-I + a + b*x)^2 + 4*b^2*(-2*a + n)*x^2*Hyp
ergeometric2F1[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)*(
I + a)^3*(-2*I + n)*x^2*(1 + I*a + I*b*x)^((I/2)*n)*(-I + a + b*x))

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (b x +a \right )}}{x^{3}}d x\]

[In]

int(exp(n*arctan(b*x+a))/x^3,x)

[Out]

int(exp(n*arctan(b*x+a))/x^3,x)

Fricas [F]

\[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=\int { \frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x^{3}} \,d x } \]

[In]

integrate(exp(n*arctan(b*x+a))/x^3,x, algorithm="fricas")

[Out]

integral(e^(n*arctan(b*x + a))/x^3, x)

Sympy [F]

\[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=\int \frac {e^{n \operatorname {atan}{\left (a + b x \right )}}}{x^{3}}\, dx \]

[In]

integrate(exp(n*atan(b*x+a))/x**3,x)

[Out]

Integral(exp(n*atan(a + b*x))/x**3, x)

Maxima [F]

\[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=\int { \frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x^{3}} \,d x } \]

[In]

integrate(exp(n*arctan(b*x+a))/x^3,x, algorithm="maxima")

[Out]

integrate(e^(n*arctan(b*x + a))/x^3, x)

Giac [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=\text {Timed out} \]

[In]

integrate(exp(n*arctan(b*x+a))/x^3,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a+b x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )}}{x^3} \,d x \]

[In]

int(exp(n*atan(a + b*x))/x^3,x)

[Out]

int(exp(n*atan(a + b*x))/x^3, x)