Integrand size = 21, antiderivative size = 124 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {144 e^{-\arctan (a x)}}{629 a c^4}-\frac {e^{-\arctan (a x)} (1-6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}-\frac {30 e^{-\arctan (a x)} (1-4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}-\frac {72 e^{-\arctan (a x)} (1-2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \] Output:
-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/exp(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)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {17 c e^{-\arctan (a x)} (-1+6 a x)-6 \left (c+a^2 c x^2\right ) \left (5 e^{-\arctan (a x)} (1-4 a x)+12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} (-i+a x) (i+a x) \left (3-2 a x+2 a^2 x^2\right )\right )}{629 a c^2 \left (c+a^2 c x^2\right )^3} \] Input:
Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^4),x]
Output:
((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)
Time = 0.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5593, 27, 5593, 5593, 5594}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^4} \, dx\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \int \frac {e^{-\arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx}{37 c}-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {30 \int \frac {e^{-\arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{37 c^4}-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \left (\frac {12}{17} \int \frac {e^{-\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )}{37 c^4}-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \left (\frac {12}{17} \left (\frac {2}{5} \int \frac {e^{-\arctan (a x)}}{a^2 x^2+1}dx-\frac {(1-2 a x) e^{-\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}\right )-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )}{37 c^4}-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5594 |
| \(\displaystyle \frac {30 \left (\frac {12}{17} \left (-\frac {(1-2 a x) e^{-\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}-\frac {2 e^{-\arctan (a x)}}{5 a}\right )-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )}{37 c^4}-\frac {(1-6 a x) e^{-\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\) |
Input:
Int[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^4),x]
Output:
-1/37*(1 - 6*a*x)/(a*c^4*E^ArcTan[a*x]*(1 + a^2*x^2)^3) + (30*(-1/17*(1 - 4*a*x)/(a*E^ArcTan[a*x]*(1 + a^2*x^2)^2) + (12*(-2/(5*a*E^ArcTan[a*x]) - ( 1 - 2*a*x)/(5*a*E^ArcTan[a*x]*(1 + a^2*x^2))))/17))/(37*c^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S imp[(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] + Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))) I nt[(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]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E ^(n*ArcTan[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
Time = 46.46 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59
| method | result | size |
| gosper | \(-\frac {\left (144 x^{6} a^{6}-144 a^{5} x^{5}+504 a^{4} x^{4}-408 a^{3} x^{3}+606 a^{2} x^{2}-366 a x +263\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\) | \(73\) |
| parallelrisch | \(\frac {\left (-144 x^{6} a^{6}+144 a^{5} x^{5}-504 a^{4} x^{4}+408 a^{3} x^{3}-606 a^{2} x^{2}+366 a x -263\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\) | \(73\) |
| orering | \(-\frac {\left (144 x^{6} a^{6}-144 a^{5} x^{5}+504 a^{4} x^{4}-408 a^{3} x^{3}+606 a^{2} x^{2}-366 a x +263\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 a \left (a^{2} c \,x^{2}+c \right )^{4}}\) | \(80\) |
Input:
int(1/exp(arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
-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
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {{\left (144 \, a^{6} x^{6} - 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} - 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} - 366 \, a x + 263\right )} e^{\left (-\arctan \left (a x\right )\right )}}{629 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")
Output:
-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)
Timed out. \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(1/exp(atan(a*x))/(a**2*c*x**2+c)**4,x)
Output:
Timed out
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")
Output:
integrate(e^(-arctan(a*x))/(a^2*c*x^2 + c)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (112) = 224\).
Time = 0.15 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.15 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {263 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} + 732 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{11} + 846 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} - 396 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} + 2313 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 2136 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} + 2372 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} - 2136 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 2313 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 396 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 846 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} - 732 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 263}{629 \, {\left (a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} + 6 \, a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 15 \, a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 20 \, a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 15 \, a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 6 \, a c^{4} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{4} e^{\left (\arctan \left (a x\right )\right )}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")
Output:
-1/629*(263*tan(1/2*arctan(a*x))^12 + 732*tan(1/2*arctan(a*x))^11 + 846*ta n(1/2*arctan(a*x))^10 - 396*tan(1/2*arctan(a*x))^9 + 2313*tan(1/2*arctan(a *x))^8 + 2136*tan(1/2*arctan(a*x))^7 + 2372*tan(1/2*arctan(a*x))^6 - 2136* tan(1/2*arctan(a*x))^5 + 2313*tan(1/2*arctan(a*x))^4 + 396*tan(1/2*arctan( a*x))^3 + 846*tan(1/2*arctan(a*x))^2 - 732*tan(1/2*arctan(a*x)) + 263)/(a* c^4*e^(arctan(a*x))*tan(1/2*arctan(a*x))^12 + 6*a*c^4*e^(arctan(a*x))*tan( 1/2*arctan(a*x))^10 + 15*a*c^4*e^(arctan(a*x))*tan(1/2*arctan(a*x))^8 + 20 *a*c^4*e^(arctan(a*x))*tan(1/2*arctan(a*x))^6 + 15*a*c^4*e^(arctan(a*x))*t an(1/2*arctan(a*x))^4 + 6*a*c^4*e^(arctan(a*x))*tan(1/2*arctan(a*x))^2 + a *c^4*e^(arctan(a*x)))
Time = 23.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {72\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {144\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {30\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x-1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \] Input:
int(exp(-atan(a*x))/(c + a^2*c*x^2)^4,x)
Output:
(72*exp(-atan(a*x))*(2*a*x - 1))/(629*a*c^4*(a^2*x^2 + 1)) - (144*exp(-ata n(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)
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {-6 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{8} x^{6}-18 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{6} x^{4}-18 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{4} x^{2}-6 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{2}-1}{e^{\mathit {atan} \left (a x \right )} a \,c^{4} \left (a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1\right )} \] Input:
int(1/exp(atan(a*x))/(a^2*c*x^2+c)^4,x)
Output:
( - 6*e**atan(a*x)*int(x/(e**atan(a*x)*a**8*x**8 + 4*e**atan(a*x)*a**6*x** 6 + 6*e**atan(a*x)*a**4*x**4 + 4*e**atan(a*x)*a**2*x**2 + e**atan(a*x)),x) *a**8*x**6 - 18*e**atan(a*x)*int(x/(e**atan(a*x)*a**8*x**8 + 4*e**atan(a*x )*a**6*x**6 + 6*e**atan(a*x)*a**4*x**4 + 4*e**atan(a*x)*a**2*x**2 + e**ata n(a*x)),x)*a**6*x**4 - 18*e**atan(a*x)*int(x/(e**atan(a*x)*a**8*x**8 + 4*e **atan(a*x)*a**6*x**6 + 6*e**atan(a*x)*a**4*x**4 + 4*e**atan(a*x)*a**2*x** 2 + e**atan(a*x)),x)*a**4*x**2 - 6*e**atan(a*x)*int(x/(e**atan(a*x)*a**8*x **8 + 4*e**atan(a*x)*a**6*x**6 + 6*e**atan(a*x)*a**4*x**4 + 4*e**atan(a*x) *a**2*x**2 + e**atan(a*x)),x)*a**2 - 1)/(e**atan(a*x)*a*c**4*(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1))