\(\int \frac {e^{-2 \arctan (a x)}}{(c+a^2 c x^2)^2} \, dx\) [292]

Optimal result

Integrand size = 21, antiderivative size = 54 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {e^{-2 \arctan (a x)}}{8 a c^2}-\frac {e^{-2 \arctan (a x)} (1-a x)}{4 a c^2 \left (1+a^2 x^2\right )} \]

[Out]

-1/8/a/c^2/exp(2*arctan(a*x))+1/4*(a*x-1)/a/c^2/exp(2*arctan(a*x))/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {(1-a x) e^{-2 \arctan (a x)}}{4 a c^2 \left (a^2 x^2+1\right )}-\frac {e^{-2 \arctan (a x)}}{8 a c^2} \]

[In]

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

[Out]

-1/8*1/(a*c^2*E^(2*ArcTan[a*x])) - (1 - a*x)/(4*a*c^2*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-a x)}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx}{4 c} \\ & = -\frac {e^{-2 \arctan (a x)}}{8 a c^2}-\frac {e^{-2 \arctan (a x)} (1-a x)}{4 a c^2 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {(1-i a x)^{-i} (1+i a x)^i \left (3-2 a x+a^2 x^2\right )}{8 c^2 \left (a+a^3 x^2\right )} \]

[In]

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

[Out]

-1/8*((1 + I*a*x)^I*(3 - 2*a*x + a^2*x^2))/(c^2*(1 - I*a*x)^I*(a + a^3*x^2))

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78

method	result	size
gosper	\(-\frac {\left (a^{2} x^{2}-2 a x +3\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{8 \left (a^{2} x^{2}+1\right ) c^{2} a}\)	\(42\)
parallelrisch	\(\frac {\left (-a^{2} x^{2}+2 a x -3\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{8 c^{2} \left (a^{2} x^{2}+1\right ) a}\)	\(43\)

[In]

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

[Out]

-1/8*(a^2*x^2-2*a*x+3)/(a^2*x^2+1)/c^2/exp(2*arctan(a*x))/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {{\left (a^{2} x^{2} - 2 \, a x + 3\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{8 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]

[In]

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

[Out]

-1/8*(a^2*x^2 - 2*a*x + 3)*e^(-2*arctan(a*x))/(a^3*c^2*x^2 + a*c^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (46) = 92\).

Time = 39.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.30 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\begin {cases} - \frac {a^{2} x^{2}}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {2 a x}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {3}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a**2*x**2/(8*a**3*c**2*x**2*exp(2*atan(a*x)) + 8*a*c**2*exp(2*atan(a*x))) + 2*a*x/(8*a**3*c**2*x**
2*exp(2*atan(a*x)) + 8*a*c**2*exp(2*atan(a*x))) - 3/(8*a**3*c**2*x**2*exp(2*atan(a*x)) + 8*a*c**2*exp(2*atan(a
*x))), Ne(a, 0)), (x/c**2, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {3}{8\,a^3\,c^2}-\frac {x}{4\,a^2\,c^2}+\frac {x^2}{8\,a\,c^2}\right )}{\frac {1}{a^2}+x^2} \]

[In]

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

[Out]

-(exp(-2*atan(a*x))*(3/(8*a^3*c^2) - x/(4*a^2*c^2) + x^2/(8*a*c^2)))/(1/a^2 + x^2)