3.337 \(\int e^{n \tan ^{-1}(a x)} (c+a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=86 \[ -\frac{c^2 2^{3-\frac{i n}{2}} (1-i a x)^{3+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-2,\frac{i n}{2}+3;\frac{i n}{2}+4;\frac{1}{2} (1-i a x)\right )}{a (-n+6 i)} \]

[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)))

________________________________________________________________________________________

Rubi [A]  time = 0.052661, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5073, 69} \[ -\frac{c^2 2^{3-\frac{i n}{2}} (1-i a x)^{3+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-2,\frac{i n}{2}+3;\frac{i n}{2}+4;\frac{1}{2} (1-i a x)\right )}{a (-n+6 i)} \]

Antiderivative was successfully verified.

[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 5073

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])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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]))

Rubi steps

\begin{align*} \int e^{n \tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^2 \, dx &=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}} \, _2F_1\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]  time = 0.0230905, size = 90, normalized size = 1.05 \[ \frac{i c^2 2^{2-\frac{i n}{2}} (1-i a x)^{3+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-2,\frac{i n}{2}+3;\frac{i n}{2}+4;\frac{1}{2} (1-i a x)\right )}{a \left (3+\frac{i n}{2}\right )} \]

Antiderivative was successfully verified.

[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]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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