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

Optimal. Leaf size=126 \[ -\frac{i a^2 \left (n^2-2\right ) e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac{i n}{2};1-\frac{i n}{2};e^{2 i \tan ^{-1}(a x)}\right )}{c n}+\frac{i a^2 \left (n^2+i n-2\right ) e^{n \tan ^{-1}(a x)}}{2 c n}-\frac{e^{n \tan ^{-1}(a x)}}{2 c x^2}-\frac{a n e^{n \tan ^{-1}(a x)}}{2 c x} \]

[Out]

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

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Rubi [A]  time = 0.182806, antiderivative size = 242, normalized size of antiderivative = 1.92, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5082, 129, 151, 155, 12, 131} \[ \frac{a^2 \left (2-n^2\right ) (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1-i a x}{i a x+1}\right )}{c (2+i n)}-\frac{a^2 \left (-i n^2+n+2 i\right ) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{a n (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^2*(2*I + n - I*n^2)*(1 - I*a*x)^((I/2)*n))/(2*c*n*(1 + I*a*x)^((I/2)*n)) - (1 - I*a*x)^((I/2)*n)/(2*c*x^2*
(1 + I*a*x)^((I/2)*n)) - (a*n*(1 - I*a*x)^((I/2)*n))/(2*c*x*(1 + I*a*x)^((I/2)*n)) + (a^2*(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 129

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^3 \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^3} \, dx}{c}\\ &=-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{\int \frac{(1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \left (-a n+2 a^2 x\right )}{x^2} \, dx}{2 c}\\ &=-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{a n (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x}+\frac{\int \frac{(1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \left (-a^2 \left (2-n^2\right )-a^3 n x\right )}{x} \, dx}{2 c}\\ &=-\frac{a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{a n (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x}-\frac{\int \frac{a^3 n \left (2-n^2\right ) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}}}{x} \, dx}{2 a c n}\\ &=-\frac{a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{a n (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x}-\frac{\left (a^2 \left (2-n^2\right )\right ) \int \frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}}}{x} \, dx}{2 c}\\ &=-\frac{a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x^2}-\frac{a n (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{2 c x}+\frac{a^2 \left (2-n^2\right ) (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 = 0.0683832, size = 174, normalized size = 1.38 \[ \frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2 a^2 n \left (n^2-2\right ) x^2 (1-i a x) \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right )+i (n-2 i) (a x-i) \left (i n \left (a^2 x^2+1\right )-2 a^2 x^2+a n^2 x (a x+i)\right )\right )}{2 c n (n-2 i) x^2 (a x-i)} \]

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

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Maple [F]  time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}}{{x}^{3} \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^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{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{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(e^(n*arctan(a*x))/((a^2*c*x^2 + c)*x^3), 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^{5} + c x^{3}}, 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(e^(n*arctan(a*x))/(a^2*c*x^5 + c*x^3), 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^{5} + x^{3}}\, 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(exp(n*atan(a*x))/(a**2*x**5 + x**3), 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^{3}}\,{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(e^(n*arctan(a*x))/((a^2*c*x^2 + c)*x^3), x)