\(\int e^{n \arctan (a x)} (c+a^2 c x^2)^2 \, dx\) [337]

Optimal result

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

[Out]

-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
)

Rubi [A] (verified)

Time = 0.04 (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 = {5181, 71} \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=-\frac {c^2 2^{3-\frac {i n}{2}} (1-i a x)^{3+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}-2,\frac {i n}{2}+3,\frac {i n}{2}+4,\frac {1}{2} (1-i a x)\right )}{a (-n+6 i)} \]

[In]

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

[Out]

-((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)))

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 5181

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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])

Rubi steps \begin{align*} \text {integral}& = c^2 \int (1-i a x)^{2+\frac {i n}{2}} (1+i a x)^{2-\frac {i n}{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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int e^{n \arctan (a x)} \left (c+a^2 c x^2\right )^2 \, dx=\frac {i 2^{2-\frac {i n}{2}} c^2 (1-i a x)^{3+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (-2+\frac {i n}{2},3+\frac {i n}{2},4+\frac {i n}{2},\frac {1}{2} (1-i a x)\right )}{a \left (3+\frac {i n}{2}\right )} \]

[In]

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

[Out]

(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))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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*ata
n(a*x)), x))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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