3.340 \(\int \frac{e^{n \tan ^{-1}(a x)} x^3}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{i \left (n^2-2\right ) e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac{i n}{2};1-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)}\right )}{a^4 c n}+\frac{\left (-i n^2+n+2 i\right ) e^{n \tan ^{-1}(a x)}}{2 a^4 c n}+\frac{x^2 e^{n \tan ^{-1}(a x)}}{2 a^2 c}-\frac{n x e^{n \tan ^{-1}(a x)}}{2 a^3 c} \]

[Out]

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

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Rubi [A]  time = 0.226768, antiderivative size = 206, normalized size of antiderivative = 1.57, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5082, 100, 143, 69} \[ \frac{2^{-1-\frac{i n}{2}} \left (2-n^2\right ) (1-i a x)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right )}{a^4 c (2+i n)}+\frac{i (1+i a x)^{-\frac{i n}{2}} \left (i a n^2 x-n^2-i n+2\right ) (1-i a x)^{\frac{i n}{2}}}{2 a^4 c n}+\frac{x^2 (1+i a x)^{-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}}}{2 a^2 c} \]

Warning: Unable to verify antiderivative.

[In]

Int[(E^(n*ArcTan[a*x])*x^3)/(c + a^2*c*x^2),x]

[Out]

(x^2*(1 - I*a*x)^((I/2)*n))/(2*a^2*c*(1 + I*a*x)^((I/2)*n)) + ((I/2)*(1 - I*a*x)^((I/2)*n)*(2 - I*n - n^2 + I*
a*n^2*x))/(a^4*c*n*(1 + I*a*x)^((I/2)*n)) + (2^(-1 - (I/2)*n)*(2 - n^2)*(1 - I*a*x)^(1 + (I/2)*n)*Hypergeometr
ic2F1[1 + (I/2)*n, 1 + (I/2)*n, 2 + (I/2)*n, (1 - I*a*x)/2])/(a^4*c*(2 + I*n))

Rule 5082

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 143

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x
)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] &&  !(SumSimplerQ[n, 1] &&  !SumSimplerQ[m, 1])

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

Mathematica [A]  time = 0.113165, size = 141, normalized size = 1.08 \[ \frac{(1-i a x)^{\frac{i n}{2}} \left (\frac{\left (a^2 n x^2+n^2 (-(a x+i))+n+2 i\right ) (1+i a x)^{-\frac{i n}{2}}}{n}+\frac{2^{-\frac{i n}{2}} \left (n^2-2\right ) (a x+i) \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right )}{n-2 i}\right )}{2 a^4 c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(n*ArcTan[a*x])*x^3)/(c + a^2*c*x^2),x]

[Out]

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

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Maple [F]  time = 0.266, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{3}}{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c),x)

[Out]

int(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

integrate(x^3*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(x^3*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} e^{n \operatorname{atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atan(a*x))*x**3/(a**2*c*x**2+c),x)

[Out]

Integral(x**3*exp(n*atan(a*x))/(a**2*x**2 + 1), x)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(x^3*e^(n*arctan(a*x))/(a^2*c*x^2 + c), x)