\(\int \frac {e^{n \arctan (a+b x)}}{x} \, dx\) [241]

Optimal result

Integrand size = 14, antiderivative size = 191 \[ \int \frac {e^{n \arctan (a+b x)}}{x} \, dx=\frac {2 i (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {i n}{2},1+\frac {i n}{2},\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a-i b x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5203, 132, 71, 12, 133} \[ \int \frac {e^{n \arctan (a+b x)}}{x} \, dx=\frac {2 i (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {i n}{2},\frac {i n}{2}+1,\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{n}-\frac {i 2^{1-\frac {i n}{2}} (-i a-i b x+1)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},\frac {i n}{2}+1,\frac {1}{2} (-i a-i b x+1)\right )}{n} \]

[In]

Int[E^(n*ArcTan[a + b*x])/x,x]

[Out]

((2*I)*(1 - I*a - I*b*x)^((I/2)*n)*Hypergeometric2F1[1, (I/2)*n, 1 + (I/2)*n, ((I - a)*(1 - I*a - I*b*x))/((I
+ a)*(1 + I*a + I*b*x))])/(n*(1 + I*a + I*b*x)^((I/2)*n)) - (I*2^(1 - (I/2)*n)*(1 - I*a - I*b*x)^((I/2)*n)*Hyp
ergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a - I*b*x)/2])/n

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 132

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.89 \[ \int \frac {e^{n \arctan (a+b x)}}{x} \, dx=\frac {2 i (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{\frac {i n}{2}} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {i n}{2},1+\frac {i n}{2},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )-2^{-\frac {i n}{2}} (1+i a+i b x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )\right )}{n} \]

[In]

Integrate[E^(n*ArcTan[a + b*x])/x,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(e^(n*arctan(b*x + a))/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(exp(n*atan(a + b*x))/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(e^(n*arctan(b*x + a))/x, x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(exp(n*atan(a + b*x))/x,x)

[Out]

int(exp(n*atan(a + b*x))/x, x)