\(\int \frac {e^{n \arctan (a x)}}{x^3 (c+a^2 c x^2)} \, dx\) [346]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 e^{n \arctan (a x)} \left (-2+i n+n^2\right )}{2 c n}-\frac {e^{n \arctan (a x)}}{2 c x^2}-\frac {a e^{n \arctan (a x)} n}{2 c x}-\frac {i a^2 e^{n \arctan (a x)} \left (-2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},e^{2 i \arctan (a x)}\right )}{c n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.85, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5190, 105, 156, 160, 12, 133} \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {i a^2 \left (2-n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},\frac {i a x+1}{1-i a x}\right )}{c n}-\frac {a^2 \left (-i n^2+n+2 i\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x} \]

[In]

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

[Out]

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

Rule 12

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

Rule 105

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, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 133

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)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 156

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, -1]

Rule 160

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 5190

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])

Rubi steps \begin{align*} \text {integral}& = \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 {i a^2 \left (2-n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i n}{2},1-\frac {i n}{2},\frac {1+i a x}{1-i a x}\right )}{c n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.38 \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (i (-2 i+n) (-i+a x) \left (-2 a^2 x^2+a n^2 x (i+a x)+i n \left (1+a^2 x^2\right )\right )+2 a^2 n \left (-2+n^2\right ) x^2 (1-i a x) \operatorname {Hypergeometric2F1}\left (1,1+\frac {i n}{2},2+\frac {i n}{2},\frac {i+a x}{i-a x}\right )\right )}{2 c n (-2 i+n) x^2 (-i+a x)} \]

[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))

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}d x\]

[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)

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]

[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

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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