\(\int \frac {e^{-2 \arctan (a x)}}{(c+a^2 c x^2)^3} \, dx\) [293]

Optimal result

Integrand size = 21, antiderivative size = 89 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 e^{-2 \arctan (a x)}}{40 a c^3}-\frac {e^{-2 \arctan (a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \arctan (a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )} \]

[Out]

-3/40/a/c^3/exp(2*arctan(a*x))+1/10*(2*a*x-1)/a/c^3/exp(2*arctan(a*x))/(a^2*x^2+1)^2-3/20*(-a*x+1)/a/c^3/exp(2
*arctan(a*x))/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 (1-a x) e^{-2 \arctan (a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac {3 e^{-2 \arctan (a x)}}{40 a c^3} \]

[In]

Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

-3/(40*a*c^3*E^(2*ArcTan[a*x])) - (1 - 2*a*x)/(10*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) - (3*(1 - a*x))/(20
*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2))

Rule 5178

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n - 2*a*(p + 1)*x)*(c + d*x^2)
^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 4*(p + 1)^2))), x] + Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))
), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 5179

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-2 \arctan (a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c} \\ & = -\frac {e^{-2 \arctan (a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \arctan (a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx}{20 c^2} \\ & = -\frac {3 e^{-2 \arctan (a x)}}{40 a c^3}-\frac {e^{-2 \arctan (a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \arctan (a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {e^{-2 \arctan (a x)} (-4+8 a x)-3 (1-i a x)^{-i} (1+i a x)^i \left (1+a^2 x^2\right ) \left (3-2 a x+a^2 x^2\right )}{40 a c^3 \left (1+a^2 x^2\right )^2} \]

[In]

Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

((-4 + 8*a*x)/E^(2*ArcTan[a*x]) - (3*(1 + I*a*x)^I*(1 + a^2*x^2)*(3 - 2*a*x + a^2*x^2))/(1 - I*a*x)^I)/(40*a*c
^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 13.84 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {\left (3 a^{4} x^{4}-6 a^{3} x^{3}+12 a^{2} x^{2}-14 a x +13\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\)	\(59\)
parallelrisch	\(\frac {\left (-3 a^{4} x^{4}+6 a^{3} x^{3}-12 a^{2} x^{2}+14 a x -13\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\)	\(59\)

[In]

int(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/40*(3*a^4*x^4-6*a^3*x^3+12*a^2*x^2-14*a*x+13)/(a^2*x^2+1)^2/c^3/exp(2*arctan(a*x))/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/40*(3*a^4*x^4 - 6*a^3*x^3 + 12*a^2*x^2 - 14*a*x + 13)*e^(-2*arctan(a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c
^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (80) = 160\).

Time = 163.45 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.55 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {3 a^{4} x^{4}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {6 a^{3} x^{3}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {12 a^{2} x^{2}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {14 a x}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {13}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-3*a**4*x**4/(40*a**5*c**3*x**4*exp(2*atan(a*x)) + 80*a**3*c**3*x**2*exp(2*atan(a*x)) + 40*a*c**3*e
xp(2*atan(a*x))) + 6*a**3*x**3/(40*a**5*c**3*x**4*exp(2*atan(a*x)) + 80*a**3*c**3*x**2*exp(2*atan(a*x)) + 40*a
*c**3*exp(2*atan(a*x))) - 12*a**2*x**2/(40*a**5*c**3*x**4*exp(2*atan(a*x)) + 80*a**3*c**3*x**2*exp(2*atan(a*x)
) + 40*a*c**3*exp(2*atan(a*x))) + 14*a*x/(40*a**5*c**3*x**4*exp(2*atan(a*x)) + 80*a**3*c**3*x**2*exp(2*atan(a*
x)) + 40*a*c**3*exp(2*atan(a*x))) - 13/(40*a**5*c**3*x**4*exp(2*atan(a*x)) + 80*a**3*c**3*x**2*exp(2*atan(a*x)
) + 40*a*c**3*exp(2*atan(a*x))), Ne(a, 0)), (x/c**3, True))

Maxima [F]

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

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]

[In]

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

[Out]

(3*exp(-2*atan(a*x))*(a*x - 1))/(20*a*c^3*(a^2*x^2 + 1)) - (3*exp(-2*atan(a*x)))/(40*a*c^3) + (exp(-2*atan(a*x
))*(2*a*x - 1))/(10*a*c^3*(a^2*x^2 + 1)^2)