Integrand size = 21, antiderivative size = 89 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {24 e^{-\arctan (a x)}}{85 a c^3}-\frac {e^{-\arctan (a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\arctan (a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \] Output:
-24/85/a/c^3/exp(arctan(a*x))-1/17*(-4*a*x+1)/a/c^3/exp(arctan(a*x))/(a^2* x^2+1)^2-12/85*(-2*a*x+1)/a/c^3/exp(arctan(a*x))/(a^2*x^2+1)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {5 e^{-\arctan (a x)} (-1+4 a x)-12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} \left (1+a^2 x^2\right ) \left (3-2 a x+2 a^2 x^2\right )}{85 a c^3 \left (1+a^2 x^2\right )^2} \] Input:
Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^3),x]
Output:
((5*(-1 + 4*a*x))/E^ArcTan[a*x] - (12*(1 + I*a*x)^(I/2)*(1 + a^2*x^2)*(3 - 2*a*x + 2*a^2*x^2))/(1 - I*a*x)^(I/2))/(85*a*c^3*(1 + a^2*x^2)^2)
Time = 0.69 (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.
\(\displaystyle \int \frac {e^{-\arctan (a x)}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {12 \int \frac {e^{-\arctan (a x)}}{c^2 \left (a^2 x^2+1\right )^2}dx}{17 c}-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {12 \int \frac {e^{-\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{17 c^3}-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 5593 |
\(\displaystyle \frac {12 \left (\frac {2}{5} \int \frac {e^{-\arctan (a x)}}{a^2 x^2+1}dx-\frac {(1-2 a x) e^{-\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}\right )}{17 c^3}-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 5594 |
\(\displaystyle \frac {12 \left (-\frac {(1-2 a x) e^{-\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}-\frac {2 e^{-\arctan (a x)}}{5 a}\right )}{17 c^3}-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}\) |
Input:
Int[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^3),x]
Output:
-1/17*(1 - 4*a*x)/(a*c^3*E^ArcTan[a*x]*(1 + a^2*x^2)^2) + (12*(-2/(5*a*E^A rcTan[a*x]) - (1 - 2*a*x)/(5*a*E^ArcTan[a*x]*(1 + a^2*x^2))))/(17*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 = 14.63 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3}+60 a^{2} x^{2}-44 a x +41\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\) | \(57\) |
parallelrisch | \(\frac {\left (-24 a^{4} x^{4}+24 a^{3} x^{3}-60 a^{2} x^{2}+44 a x -41\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(57\) |
orering | \(-\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3}+60 a^{2} x^{2}-44 a x +41\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 a \left (a^{2} c \,x^{2}+c \right )^{3}}\) | \(64\) |
Input:
int(1/exp(arctan(a*x))/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/85*(24*a^4*x^4-24*a^3*x^3+60*a^2*x^2-44*a*x+41)/(a^2*x^2+1)^2/c^3/exp(a rctan(a*x))/a
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {{\left (24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 44 \, a x + 41\right )} e^{\left (-\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")
Output:
-1/85*(24*a^4*x^4 - 24*a^3*x^3 + 60*a^2*x^2 - 44*a*x + 41)*e^(-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. 291 vs. \(2 (78) = 156\).
Time = 90.23 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.27 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {24 a^{4} x^{4}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {24 a^{3} x^{3}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {60 a^{2} x^{2}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {44 a x}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {41}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(1/exp(atan(a*x))/(a**2*c*x**2+c)**3,x)
Output:
Piecewise((-24*a**4*x**4/(85*a**5*c**3*x**4*exp(atan(a*x)) + 170*a**3*c**3 *x**2*exp(atan(a*x)) + 85*a*c**3*exp(atan(a*x))) + 24*a**3*x**3/(85*a**5*c **3*x**4*exp(atan(a*x)) + 170*a**3*c**3*x**2*exp(atan(a*x)) + 85*a*c**3*ex p(atan(a*x))) - 60*a**2*x**2/(85*a**5*c**3*x**4*exp(atan(a*x)) + 170*a**3* c**3*x**2*exp(atan(a*x)) + 85*a*c**3*exp(atan(a*x))) + 44*a*x/(85*a**5*c** 3*x**4*exp(atan(a*x)) + 170*a**3*c**3*x**2*exp(atan(a*x)) + 85*a*c**3*exp( atan(a*x))) - 41/(85*a**5*c**3*x**4*exp(atan(a*x)) + 170*a**3*c**3*x**2*ex p(atan(a*x)) + 85*a*c**3*exp(atan(a*x))), Ne(a, 0)), (x/c**3, True))
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")
Output:
integrate(e^(-arctan(a*x))/(a^2*c*x^2 + c)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (80) = 160\).
Time = 0.14 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {41 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 88 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} + 76 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} - 72 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 150 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 72 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 76 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} - 88 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 41}{85 \, {\left (a c^{3} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 4 \, a c^{3} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 6 \, a c^{3} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 4 \, a c^{3} e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{3} e^{\left (\arctan \left (a x\right )\right )}\right )}} \] Input:
integrate(1/exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
-1/85*(41*tan(1/2*arctan(a*x))^8 + 88*tan(1/2*arctan(a*x))^7 + 76*tan(1/2* arctan(a*x))^6 - 72*tan(1/2*arctan(a*x))^5 + 150*tan(1/2*arctan(a*x))^4 + 72*tan(1/2*arctan(a*x))^3 + 76*tan(1/2*arctan(a*x))^2 - 88*tan(1/2*arctan( a*x)) + 41)/(a*c^3*e^(arctan(a*x))*tan(1/2*arctan(a*x))^8 + 4*a*c^3*e^(arc tan(a*x))*tan(1/2*arctan(a*x))^6 + 6*a*c^3*e^(arctan(a*x))*tan(1/2*arctan( a*x))^4 + 4*a*c^3*e^(arctan(a*x))*tan(1/2*arctan(a*x))^2 + a*c^3*e^(arctan (a*x)))
Time = 23.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {12\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {24\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \] Input:
int(exp(-atan(a*x))/(c + a^2*c*x^2)^3,x)
Output:
(12*exp(-atan(a*x))*(2*a*x - 1))/(85*a*c^3*(a^2*x^2 + 1)) - (24*exp(-atan( a*x)))/(85*a*c^3) + (exp(-atan(a*x))*(4*a*x - 1))/(17*a*c^3*(a^2*x^2 + 1)^ 2)
\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-4 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{6} x^{4}-8 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{4} x^{2}-4 e^{\mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{\mathit {atan} \left (a x \right )} a^{6} x^{6}+3 e^{\mathit {atan} \left (a x \right )} a^{4} x^{4}+3 e^{\mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{\mathit {atan} \left (a x \right )}}d x \right ) a^{2}-1}{e^{\mathit {atan} \left (a x \right )} a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int(1/exp(atan(a*x))/(a^2*c*x^2+c)^3,x)
Output:
( - 4*e**atan(a*x)*int(x/(e**atan(a*x)*a**6*x**6 + 3*e**atan(a*x)*a**4*x** 4 + 3*e**atan(a*x)*a**2*x**2 + e**atan(a*x)),x)*a**6*x**4 - 8*e**atan(a*x) *int(x/(e**atan(a*x)*a**6*x**6 + 3*e**atan(a*x)*a**4*x**4 + 3*e**atan(a*x) *a**2*x**2 + e**atan(a*x)),x)*a**4*x**2 - 4*e**atan(a*x)*int(x/(e**atan(a* x)*a**6*x**6 + 3*e**atan(a*x)*a**4*x**4 + 3*e**atan(a*x)*a**2*x**2 + e**at an(a*x)),x)*a**2 - 1)/(e**atan(a*x)*a*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))