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

Optimal. Leaf size=84 \[ -\frac{c 2^{2-\frac{i n}{2}} (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-1,\frac{i n}{2}+2;\frac{i n}{2}+3;\frac{1}{2} (1-i a x)\right )}{a (-n+4 i)} \]

[Out]

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

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Rubi [A]  time = 0.0390547, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5073, 69} \[ -\frac{c 2^{2-\frac{i n}{2}} (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-1,\frac{i n}{2}+2;\frac{i n}{2}+3;\frac{1}{2} (1-i a x)\right )}{a (-n+4 i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(2 - (I/2)*n)*c*(1 - I*a*x)^(2 + (I/2)*n)*Hypergeometric2F1[-1 + (I/2)*n, 2 + (I/2)*n, 3 + (I/2)*n, (1 -
I*a*x)/2])/(a*(4*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 ) \, dx &=c \int (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{1-\frac{i n}{2}} \, dx\\ &=-\frac{2^{2-\frac{i n}{2}} c (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left (-1+\frac{i n}{2},2+\frac{i n}{2};3+\frac{i n}{2};\frac{1}{2} (1-i a x)\right )}{a (4 i-n)}\\ \end{align*}

Mathematica [A]  time = 0.0155312, size = 88, normalized size = 1.05 \[ \frac{i c 2^{1-\frac{i n}{2}} (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}-1,\frac{i n}{2}+2;\frac{i n}{2}+3;\frac{1}{2} (1-i a x)\right )}{a \left (2+\frac{i n}{2}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }} \left ({a}^{2}c{x}^{2}+c \right ) \, 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),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} 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),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{2} + c\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),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int a^{2} x^{2} 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),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} 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),x, algorithm="giac")

[Out]

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