Integrand size = 21, antiderivative size = 89 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 e^{-2 \arctan (a x)}}{40 a c^3}-\frac {e^{-2 \arctan (a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \arctan (a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )} \] Output:
-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)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {e^{-2 \arctan (a x)} (-4+8 a x)-3 (1-i a x)^{-i} (1+i a x)^i \left (1+a^2 x^2\right ) \left (3-2 a x+a^2 x^2\right )}{40 a c^3 \left (1+a^2 x^2\right )^2} \] Input:
Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]
Output:
((-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)
Time = 0.67 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5593, 27, 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.
\(\displaystyle \int \frac {e^{-2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {3 \int \frac {e^{-2 \arctan (a x)}}{c^2 \left (a^2 x^2+1\right )^2}dx}{5 c}-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \int \frac {e^{-2 \arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{5 c^3}-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {3 \left (\frac {1}{4} \int \frac {e^{-2 \arctan (a x)}}{a^2 x^2+1}dx-\frac {(1-a x) e^{-2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}\right )}{5 c^3}-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 5594 |
\(\displaystyle \frac {3 \left (-\frac {(1-a x) e^{-2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}-\frac {e^{-2 \arctan (a x)}}{8 a}\right )}{5 c^3}-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}\) |
Input:
Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]
Output:
-1/10*(1 - 2*a*x)/(a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) + (3*(-1/8*1/( a*E^(2*ArcTan[a*x])) - (1 - a*x)/(4*a*E^(2*ArcTan[a*x])*(1 + a^2*x^2))))/( 5*c^3)
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 = 14.87 (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 | \(\frac {\left (-3 a^{4} x^{4}+6 a^{3} x^{3}-12 a^{2} x^{2}+14 a x -13\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(59\) |
orering | \(-\frac {\left (3 a^{4} x^{4}-6 a^{3} x^{3}+12 a^{2} x^{2}-14 a x +13\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 a \left (a^{2} c \,x^{2}+c \right )^{3}}\) | \(66\) |
Input:
int(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
-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*a rctan(a*x))/a
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \] Input:
integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (82) = 164\).
Time = 173.99 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.55 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {3 a^{4} x^{4}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {6 a^{3} x^{3}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {12 a^{2} x^{2}}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {14 a x}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {13}{40 a^{5} c^{3} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}} + 80 a^{3} c^{3} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 40 a c^{3} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**3,x)
Output:
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*exp(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)) + 8 0*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))
\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:
integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")
Output:
integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (79) = 158\).
Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.16 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {13 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 28 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} - 4 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} - 36 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 30 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 36 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} - 4 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} - 28 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 13}{40 \, {\left (a c^{3} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 4 \, a c^{3} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 6 \, a c^{3} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 4 \, a c^{3} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{3} e^{\left (2 \, \arctan \left (a x\right )\right )}\right )}} \] Input:
integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
-1/40*(13*tan(1/2*arctan(a*x))^8 + 28*tan(1/2*arctan(a*x))^7 - 4*tan(1/2*a rctan(a*x))^6 - 36*tan(1/2*arctan(a*x))^5 + 30*tan(1/2*arctan(a*x))^4 + 36 *tan(1/2*arctan(a*x))^3 - 4*tan(1/2*arctan(a*x))^2 - 28*tan(1/2*arctan(a*x )) + 13)/(a*c^3*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^8 + 4*a*c^3*e^(2*ar ctan(a*x))*tan(1/2*arctan(a*x))^6 + 6*a*c^3*e^(2*arctan(a*x))*tan(1/2*arct an(a*x))^4 + 4*a*c^3*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^2 + a*c^3*e^(2 *arctan(a*x)))
Time = 23.90 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \] Input:
int(exp(-2*atan(a*x))/(c + a^2*c*x^2)^3,x)
Output:
(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)
\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-4 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{6} x^{4}-8 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{4} x^{2}-4 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{2}-1}{2 e^{2 \mathit {atan} \left (a x \right )} a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int(1/exp(2*atan(a*x))/(a^2*c*x^2+c)^3,x)
Output:
( - 4*e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))*a**6*x**6 + 3*e**(2*atan(a* x))*a**4*x**4 + 3*e**(2*atan(a*x))*a**2*x**2 + e**(2*atan(a*x))),x)*a**6*x **4 - 8*e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))*a**6*x**6 + 3*e**(2*atan( a*x))*a**4*x**4 + 3*e**(2*atan(a*x))*a**2*x**2 + e**(2*atan(a*x))),x)*a**4 *x**2 - 4*e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))*a**6*x**6 + 3*e**(2*ata n(a*x))*a**4*x**4 + 3*e**(2*atan(a*x))*a**2*x**2 + e**(2*atan(a*x))),x)*a* *2 - 1)/(2*e**(2*atan(a*x))*a*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))