\(\int \frac {e^{-2 \arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [294]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {9 e^{-2 \arctan (a x)}}{160 a c^4}-\frac {e^{-2 \arctan (a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \arctan (a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \arctan (a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )} \]

[Out]

-9/160/a/c^4/exp(2*arctan(a*x))+1/20*(3*a*x-1)/a/c^4/exp(2*arctan(a*x))/(a^2*x^2+1)^3-3/40*(-2*a*x+1)/a/c^4/ex
p(2*arctan(a*x))/(a^2*x^2+1)^2-9/80*(-a*x+1)/a/c^4/exp(2*arctan(a*x))/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}-\frac {9 (1-a x) e^{-2 \arctan (a x)}}{80 a c^4 \left (a^2 x^2+1\right )}-\frac {3 (1-2 a x) e^{-2 \arctan (a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}-\frac {9 e^{-2 \arctan (a x)}}{160 a c^4} \]

[In]

Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^4),x]

[Out]

-9/(160*a*c^4*E^(2*ArcTan[a*x])) - (1 - 3*a*x)/(20*a*c^4*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^3) - (3*(1 - 2*a*x))/
(40*a*c^4*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) - (9*(1 - a*x))/(80*a*c^4*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)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{4 c} \\ & = -\frac {e^{-2 \arctan (a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \arctan (a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{20 c^2} \\ & = -\frac {e^{-2 \arctan (a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \arctan (a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \arctan (a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )}+\frac {9 \int \frac {e^{-2 \arctan (a x)}}{c+a^2 c x^2} \, dx}{80 c^3} \\ & = -\frac {9 e^{-2 \arctan (a x)}}{160 a c^4}-\frac {e^{-2 \arctan (a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \arctan (a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \arctan (a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {8 c e^{-2 \arctan (a x)} (-1+3 a x)-3 \left (c+a^2 c x^2\right ) \left (e^{-2 \arctan (a x)} (4-8 a x)+3 (1-i a x)^{-i} (1+i a x)^i (-i+a x) (i+a x) \left (3-2 a x+a^2 x^2\right )\right )}{160 a c^2 \left (c+a^2 c x^2\right )^3} \]

[In]

Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^4),x]

[Out]

((8*c*(-1 + 3*a*x))/E^(2*ArcTan[a*x]) - 3*(c + a^2*c*x^2)*((4 - 8*a*x)/E^(2*ArcTan[a*x]) + (3*(1 + I*a*x)^I*(-
I + a*x)*(I + a*x)*(3 - 2*a*x + a^2*x^2))/(1 - I*a*x)^I))/(160*a*c^2*(c + a^2*c*x^2)^3)

Maple [A] (verified)

Time = 42.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {\left (9 a^{6} x^{6}-18 a^{5} x^{5}+45 a^{4} x^{4}-60 a^{3} x^{3}+75 a^{2} x^{2}-66 a x +47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(75\)
parallelrisch	\(\frac {\left (-9 a^{6} x^{6}+18 a^{5} x^{5}-45 a^{4} x^{4}+60 a^{3} x^{3}-75 a^{2} x^{2}+66 a x -47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(75\)

[In]

int(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)

[Out]

-1/160*(9*a^6*x^6-18*a^5*x^5+45*a^4*x^4-60*a^3*x^3+75*a^2*x^2-66*a*x+47)/(a^2*x^2+1)^3/c^4/exp(2*arctan(a*x))/
a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {{\left (9 \, a^{6} x^{6} - 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} - 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} - 66 \, a x + 47\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-1/160*(9*a^6*x^6 - 18*a^5*x^5 + 45*a^4*x^4 - 60*a^3*x^3 + 75*a^2*x^2 - 66*a*x + 47)*e^(-2*arctan(a*x))/(a^7*c
^4*x^6 + 3*a^5*c^4*x^4 + 3*a^3*c^4*x^2 + a*c^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \]

[In]

integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^4, x)

Giac [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x-1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]

[In]

int(exp(-2*atan(a*x))/(c + a^2*c*x^2)^4,x)

[Out]

(9*exp(-2*atan(a*x))*(a*x - 1))/(80*a*c^4*(a^2*x^2 + 1)) - (9*exp(-2*atan(a*x)))/(160*a*c^4) + (3*exp(-2*atan(
a*x))*(2*a*x - 1))/(40*a*c^4*(a^2*x^2 + 1)^2) + (exp(-2*atan(a*x))*(3*a*x - 1))/(20*a*c^4*(a^2*x^2 + 1)^3)