\(\int \frac {e^{n \arctan (a x)}}{x^3 (c+a^2 c x^2)} \, dx\) [360]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 e^{n \arctan (a x)} \left (-2+i n+n^2\right )}{2 c n}-\frac {e^{n \arctan (a x)}}{2 c x^2}-\frac {a e^{n \arctan (a x)} n}{2 c x}-\frac {i a^2 e^{n \arctan (a x)} \left (-2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},e^{2 i \arctan (a x)}\right )}{c n} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.38 \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (i (-2 i+n) (-i+a x) \left (-2 a^2 x^2+a n^2 x (i+a x)+i n \left (1+a^2 x^2\right )\right )+2 a^2 n \left (-2+n^2\right ) x^2 (1-i a x) \operatorname {Hypergeometric2F1}\left (1,1+\frac {i n}{2},2+\frac {i n}{2},\frac {i+a x}{i-a x}\right )\right )}{2 c n (-2 i+n) x^2 (-i+a x)} \] Input:

Integrate[E^(n*ArcTan[a*x])/(x^3*(c + a^2*c*x^2)),x]
 

Output:

((1 - I*a*x)^((I/2)*n)*(I*(-2*I + n)*(-I + a*x)*(-2*a^2*x^2 + a*n^2*x*(I + 
 a*x) + I*n*(1 + a^2*x^2)) + 2*a^2*n*(-2 + n^2)*x^2*(1 - I*a*x)*Hypergeome 
tric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (I + a*x)/(I - a*x)]))/(2*c*n*(-2*I + 
 n)*x^2*(1 + I*a*x)^((I/2)*n)*(-I + a*x))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.77, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5605, 114, 25, 27, 168, 27, 172, 25, 27, 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.

\(\displaystyle \int \frac {e^{n \arctan (a x)}}{x^3 \left (a^2 c x^2+c\right )} \, dx\)

\(\Big \downarrow \) 5605

\(\displaystyle \frac {\int \frac {(1-i a x)^{\frac {i n}{2}-1} (i a x+1)^{-\frac {i n}{2}-1}}{x^3}dx}{c}\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

Int[E^(n*ArcTan[a*x])/(x^3*(c + a^2*c*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(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] + Simp[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*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

rule 5605

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^p   Int[x^m*(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I* 
(n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Integer 
Q[p] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}d x\]

Input:

int(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c),x)
 

Output:

int(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c),x)
 

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

integral(e^(n*arctan(a*x))/(a^2*c*x^5 + c*x^3), x)
 

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{5} + x^{3}}\, dx}{c} \] Input:

integrate(exp(n*atan(a*x))/x**3/(a**2*c*x**2+c),x)
 

Output:

Integral(exp(n*atan(a*x))/(a**2*x**5 + x**3), x)/c
 

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

integrate(e^(n*arctan(a*x))/((a^2*c*x^2 + c)*x^3), x)
 

Giac [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c),x, algorithm="giac")
 

Output:

integrate(e^(n*arctan(a*x))/((a^2*c*x^2 + c)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \] Input:

int(exp(n*atan(a*x))/(x^3*(c + a^2*c*x^2)),x)
 

Output:

int(exp(n*atan(a*x))/(x^3*(c + a^2*c*x^2)), x)
 

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {-e^{\mathit {atan} \left (a x \right ) n} a^{2} x^{2}-e^{\mathit {atan} \left (a x \right ) n} a n x -e^{\mathit {atan} \left (a x \right ) n}+\left (\int \frac {e^{\mathit {atan} \left (a x \right ) n}}{a^{2} x^{3}+x}d x \right ) a^{2} n^{2} x^{2}-2 \left (\int \frac {e^{\mathit {atan} \left (a x \right ) n}}{a^{2} x^{3}+x}d x \right ) a^{2} x^{2}}{2 c \,x^{2}} \] Input:

int(exp(n*atan(a*x))/x^3/(a^2*c*x^2+c),x)
 

Output:

( - e**(atan(a*x)*n)*a**2*x**2 - e**(atan(a*x)*n)*a*n*x - e**(atan(a*x)*n) 
 + int(e**(atan(a*x)*n)/(a**2*x**3 + x),x)*a**2*n**2*x**2 - 2*int(e**(atan 
(a*x)*n)/(a**2*x**3 + x),x)*a**2*x**2)/(2*c*x**2)