\(\int \frac {e^{-\arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [280]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {144 e^{-\arctan (a x)}}{629 a c^4}-\frac {e^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}-\frac {30 e^{-\arctan (a x)} (1-4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}-\frac {72 e^{-\arctan (a x)} (1-2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \]

[Out]

-144/629/a/c^4/exp(arctan(a*x))+1/37*(6*a*x-1)/a/c^4/exp(arctan(a*x))/(a^2*x^2+1)^3-30/629*(-4*a*x+1)/a/c^4/ex
p(arctan(a*x))/(a^2*x^2+1)^2-72/629*(-2*a*x+1)/a/c^4/exp(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} \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}-\frac {72 (1-2 a x) e^{-\arctan (a x)}}{629 a c^4 \left (a^2 x^2+1\right )}-\frac {30 (1-4 a x) e^{-\arctan (a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}-\frac {144 e^{-\arctan (a x)}}{629 a c^4} \]

[In]

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

[Out]

-144/(629*a*c^4*E^ArcTan[a*x]) - (1 - 6*a*x)/(37*a*c^4*E^ArcTan[a*x]*(1 + a^2*x^2)^3) - (30*(1 - 4*a*x))/(629*
a*c^4*E^ArcTan[a*x]*(1 + a^2*x^2)^2) - (72*(1 - 2*a*x))/(629*a*c^4*E^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^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{37 c} \\ & = -\frac {e^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}-\frac {30 e^{-\arctan (a x)} (1-4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {360 \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{629 c^2} \\ & = -\frac {e^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}-\frac {30 e^{-\arctan (a x)} (1-4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}-\frac {72 e^{-\arctan (a x)} (1-2 a x)}{629 a c^4 \left (1+a^2 x^2\right )}+\frac {144 \int \frac {e^{-\arctan (a x)}}{c+a^2 c x^2} \, dx}{629 c^3} \\ & = -\frac {144 e^{-\arctan (a x)}}{629 a c^4}-\frac {e^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}-\frac {30 e^{-\arctan (a x)} (1-4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}-\frac {72 e^{-\arctan (a x)} (1-2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {17 c e^{-\arctan (a x)} (-1+6 a x)-6 \left (c+a^2 c x^2\right ) \left (5 e^{-\arctan (a x)} (1-4 a x)+12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} (-i+a x) (i+a x) \left (3-2 a x+2 a^2 x^2\right )\right )}{629 a c^2 \left (c+a^2 c x^2\right )^3} \]

[In]

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

[Out]

((17*c*(-1 + 6*a*x))/E^ArcTan[a*x] - 6*(c + a^2*c*x^2)*((5*(1 - 4*a*x))/E^ArcTan[a*x] + (12*(1 + I*a*x)^(I/2)*
(-I + a*x)*(I + a*x)*(3 - 2*a*x + 2*a^2*x^2))/(1 - I*a*x)^(I/2)))/(629*a*c^2*(c + a^2*c*x^2)^3)

Maple [A] (verified)

Time = 40.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59

method	result	size
gosper	\(-\frac {\left (144 a^{6} x^{6}-144 a^{5} x^{5}+504 a^{4} x^{4}-408 a^{3} x^{3}+606 a^{2} x^{2}-366 a x +263\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(73\)
parallelrisch	\(\frac {\left (-144 a^{6} x^{6}+144 a^{5} x^{5}-504 a^{4} x^{4}+408 a^{3} x^{3}-606 a^{2} x^{2}+366 a x -263\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(73\)

[In]

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

[Out]

-1/629*(144*a^6*x^6-144*a^5*x^5+504*a^4*x^4-408*a^3*x^3+606*a^2*x^2-366*a*x+263)/(a^2*x^2+1)^3/c^4/exp(arctan(
a*x))/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {{\left (144 \, a^{6} x^{6} - 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} - 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} - 366 \, a x + 263\right )} e^{\left (-\arctan \left (a x\right )\right )}}{629 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]

[In]

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

[Out]

-1/629*(144*a^6*x^6 - 144*a^5*x^5 + 504*a^4*x^4 - 408*a^3*x^3 + 606*a^2*x^2 - 366*a*x + 263)*e^(-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. \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate(1/exp(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 = 112, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {72\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {144\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {30\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x-1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]

[In]

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

[Out]

(72*exp(-atan(a*x))*(2*a*x - 1))/(629*a*c^4*(a^2*x^2 + 1)) - (144*exp(-atan(a*x)))/(629*a*c^4) + (30*exp(-atan
(a*x))*(4*a*x - 1))/(629*a*c^4*(a^2*x^2 + 1)^2) + (exp(-atan(a*x))*(6*a*x - 1))/(37*a*c^4*(a^2*x^2 + 1)^3)