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^(3-1/2*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
)

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Rubi [A]  time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 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]))

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])

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*}

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Mathematica [A]  time = 0.02, 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))

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{2}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{2} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int 2 a^{2} x^{2} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

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))

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