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

Optimal. Leaf size=65 \[ \frac{i e^{n \tan ^{-1}(a x)}}{c n}-\frac{2 i e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac{i n}{2};1-\frac{i n}{2};e^{2 i \tan ^{-1}(a x)}\right )}{c n} \]

[Out]

(I*E^(n*ArcTan[a*x]))/(c*n) - ((2*I)*E^(n*ArcTan[a*x])*Hypergeometric2F1[1, (-I/2)*n, 1 - (I/2)*n, E^((2*I)*Ar
cTan[a*x])])/(c*n)

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Rubi [B]  time = 0.0984313, antiderivative size = 132, normalized size of antiderivative = 2.03, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5082, 96, 131} \[ \frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c n}-\frac{2 (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1-i a x}{i a x+1}\right )}{c (2+i n)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(I*(1 - I*a*x)^((I/2)*n))/(c*n*(1 + I*a*x)^((I/2)*n)) - (2*(1 - I*a*x)^(1 + (I/2)*n)*(1 + I*a*x)^(-1 - (I/2)*n
)*Hypergeometric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (1 - I*a*x)/(1 + I*a*x)])/(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 96

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.0363659, size = 120, normalized size = 1.85 \[ \frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2 (n-i a n x) \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right )+(2+i n) (a x-i)\right )}{c n (n-2 i) (a x-i)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}}{x \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))/x/(a^2*c*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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