\(\int \frac {e^{n \arctan (a x)} x^3}{c+a^2 c x^2} \, dx\) [354]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {e^{n \arctan (a x)} x^3}{c+a^2 c x^2} \, dx=\frac {e^{n \arctan (a x)} \left (2 i+n-i n^2\right )}{2 a^4 c n}-\frac {e^{n \arctan (a x)} n x}{2 a^3 c}+\frac {e^{n \arctan (a x)} x^2}{2 a^2 c}+\frac {i e^{n \arctan (a x)} \left (-2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},-e^{2 i \arctan (a x)}\right )}{a^4 c n} \] Output:

1/2*exp(n*arctan(a*x))*(2*I+n-I*n^2)/a^4/c/n-1/2*exp(n*arctan(a*x))*n*x/a^ 
3/c+1/2*exp(n*arctan(a*x))*x^2/a^2/c+I*exp(n*arctan(a*x))*(n^2-2)*hypergeo 
m([1, -1/2*I*n],[1-1/2*I*n],-(1+I*a*x)^2/(a^2*x^2+1))/a^4/c/n
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08 \[ \int \frac {e^{n \arctan (a x)} x^3}{c+a^2 c x^2} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} \left (\frac {(1+i a x)^{-\frac {i n}{2}} \left (2 i+n+a^2 n x^2-n^2 (i+a x)\right )}{n}+\frac {2^{-\frac {i n}{2}} \left (-2+n^2\right ) (i+a x) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},1+\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{-2 i+n}\right )}{2 a^4 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.59, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5605, 111, 25, 160, 79}

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.

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

\(\Big \downarrow \) 5605

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 160

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 111

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 160

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

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

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]

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

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

Output:

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

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{3} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \] Input:

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

Output:

Integral(x**3*exp(n*atan(a*x))/(a**2*x**2 + 1), x)/c
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {e^{\mathit {atan} \left (a x \right ) n} x^{3}}{a^{2} x^{2}+1}d x}{c} \] Input:

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

Output:

int((e**(atan(a*x)*n)*x**3)/(a**2*x**2 + 1),x)/c