\(\int \frac {e^{n \arctan (a+b x)}}{x} \, dx\) [255]

Optimal result

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

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
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.89 \[ \int \frac {e^{n \arctan (a+b x)}}{x} \, dx=\frac {2 i (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{\frac {i n}{2}} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {i n}{2},1+\frac {i n}{2},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )-2^{-\frac {i n}{2}} (1+i a+i b x)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},1+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )\right )}{n} \] Input:

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

Output:

((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))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5618, 140, 27, 79, 141}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5618

\(\displaystyle \int \frac {(-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}}}{x}dx\)

\(\Big \downarrow \) 140

\(\displaystyle \int \frac {(1-i a) (-i a-i b x+1)^{\frac {i n}{2}-1} (i a+i b x+1)^{-\frac {i n}{2}}}{x}dx-i b \int (-i a-i b x+1)^{\frac {i n}{2}-1} (i a+i b x+1)^{-\frac {i n}{2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle (1-i a) \int \frac {(-i a-i b x+1)^{\frac {i n}{2}-1} (i a+i b x+1)^{-\frac {i n}{2}}}{x}dx-i b \int (-i a-i b x+1)^{\frac {i n}{2}-1} (i a+i b x+1)^{-\frac {i n}{2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle (1-i a) \int \frac {(-i a-i b x+1)^{\frac {i n}{2}-1} (i a+i b x+1)^{-\frac {i n}{2}}}{x}dx-\frac {i 2^{1-\frac {i n}{2}} (-i a-i b x+1)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},\frac {i n}{2}+1,\frac {1}{2} (-i a-i b x+1)\right )}{n}\)

\(\Big \downarrow \) 141

\(\displaystyle -\frac {2 (1-i a) (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {i n}{2},\frac {i n}{2}+1,\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(a+i) n}-\frac {i 2^{1-\frac {i n}{2}} (-i a-i b x+1)^{\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2},\frac {i n}{2},\frac {i n}{2}+1,\frac {1}{2} (-i a-i b x+1)\right )}{n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

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] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[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)*ExpandToSum[(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 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 5618

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atan(a + b*x)*n)/x,x)