\(\int e^{n \arctan (a x)} (c+a^2 c x^2)^2 \, dx\) [351]

Optimal result

Integrand size = 21, antiderivative size = 86 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=-\frac {2^{3-\frac {i n}{2}} c^2 (1-i a x)^{3+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (-2+\frac {i n}{2},3+\frac {i n}{2},4+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a (6 i-n)} \] Output:

-2^(3-1/2*I*n)*c^2*(1-I*a*x)^(3+1/2*I*n)*hypergeom([-2+1/2*I*n, 3+1/2*I*n] 
,[4+1/2*I*n],1/2-1/2*I*a*x)/a/(6*I-n)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=\frac {i 2^{2-\frac {i n}{2}} c^2 (1-i a x)^{3+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (-2+\frac {i n}{2},3+\frac {i n}{2},4+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a \left (3+\frac {i n}{2}\right )} \] Input:

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

Output:

(I*2^(2 - (I/2)*n)*c^2*(1 - I*a*x)^(3 + (I/2)*n)*Hypergeometric2F1[-2 + (I 
/2)*n, 3 + (I/2)*n, 4 + (I/2)*n, (1 - I*a*x)/2])/(a*(3 + (I/2)*n))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5596, 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.

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

\(\Big \downarrow \) 5596

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {c^2 2^{3-\frac {i n}{2}} (1-i a x)^{3+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-2,\frac {i n}{2}+3,\frac {i n}{2}+4,\frac {1}{2} (1-i a x)\right )}{a (-n+6 i)}\)

Input:

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

Output:

-((2^(3 - (I/2)*n)*c^2*(1 - I*a*x)^(3 + (I/2)*n)*Hypergeometric2F1[-2 + (I 
/2)*n, 3 + (I/2)*n, 4 + (I/2)*n, (1 - I*a*x)/2])/(a*(6*I - n)))
 

Defintions of rubi rules used

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 5596

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*e^(n*arctan(a*x)), x)
 

Sympy [F]

\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\int 2 a^{2} x^{2} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \] Input:

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

Output:

c**2*(Integral(2*a**2*x**2*exp(n*atan(a*x)), x) + Integral(a**4*x**4*exp(n 
*atan(a*x)), x) + Integral(exp(n*atan(a*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=c^{2} \left (\int e^{\mathit {atan} \left (a x \right ) n}d x +\left (\int e^{\mathit {atan} \left (a x \right ) n} x^{4}d x \right ) a^{4}+2 \left (\int e^{\mathit {atan} \left (a x \right ) n} x^{2}d x \right ) a^{2}\right ) \] Input:

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

Output:

c**2*(int(e**(atan(a*x)*n),x) + int(e**(atan(a*x)*n)*x**4,x)*a**4 + 2*int( 
e**(atan(a*x)*n)*x**2,x)*a**2)