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

Optimal. Leaf size=63 \[ -\frac{\left (\frac{1}{37}-\frac{6 i}{37}\right ) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left (-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a} \]

[Out]

((-1/37 + (6*I)/37)*2^(3 + I/2)*c^2*(1 - I*a*x)^(3 - I/2)*Hypergeometric2F1[-2 - I/2, 3 - I/2, 4 - I/2, (1 - I
*a*x)/2])/a

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Rubi [A]  time = 0.0366298, antiderivative size = 63, 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} \[ -\frac{\left (\frac{1}{37}-\frac{6 i}{37}\right ) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left (-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/37 + (6*I)/37)*2^(3 + I/2)*c^2*(1 - I*a*x)^(3 - I/2)*Hypergeometric2F1[-2 - I/2, 3 - I/2, 4 - I/2, (1 - I
*a*x)/2])/a

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^{-\tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^2 \, dx &=c^2 \int (1-i a x)^{2-\frac{i}{2}} (1+i a x)^{2+\frac{i}{2}} \, dx\\ &=-\frac{\left (\frac{1}{37}-\frac{6 i}{37}\right ) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left (-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0138363, size = 63, normalized size = 1. \[ -\frac{\left (\frac{1}{37}-\frac{6 i}{37}\right ) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left (-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/37 + (6*I)/37)*2^(3 + I/2)*c^2*(1 - I*a*x)^(3 - I/2)*Hypergeometric2F1[-2 - I/2, 3 - I/2, 4 - I/2, (1 - I
*a*x)/2])/a

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{{{\rm e}^{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*c*x^2+c)^2/exp(arctan(a*x)),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 )}^{2} e^{\left (-\arctan \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 (-\arctan \left (a x\right )\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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