Integrand size = 21, antiderivative size = 181 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {720 e^{n \arctan (a x)}}{a c^4 n \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )}+\frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 e^{n \arctan (a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac {360 e^{n \arctan (a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )} \] Output:
720*exp(n*arctan(a*x))/a/c^4/n/(n^2+4)/(n^2+16)/(n^2+36)+exp(n*arctan(a*x) )*(6*a*x+n)/a/c^4/(n^2+36)/(a^2*x^2+1)^3+30*exp(n*arctan(a*x))*(4*a*x+n)/a /c^4/(n^2+16)/(n^2+36)/(a^2*x^2+1)^2+360*exp(n*arctan(a*x))*(2*a*x+n)/a/c^ 4/(n^2+4)/(n^2+16)/(n^2+36)/(a^2*x^2+1)
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.91 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {e^{n \arctan (a x)} (n+6 a x)+\frac {30 \left (c+a^2 c x^2\right ) \left (e^{n \arctan (a x)} n (-2 i+n) (2 i+n) (n+4 a x)+12 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} (-i+a x) (i+a x) \left (2+n^2+2 a n x+2 a^2 x^2\right )\right )}{c n \left (64+20 n^2+n^4\right )}}{a c \left (36+n^2\right ) \left (c+a^2 c x^2\right )^3} \] Input:
Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]
Output:
(E^(n*ArcTan[a*x])*(n + 6*a*x) + (30*(c + a^2*c*x^2)*(E^(n*ArcTan[a*x])*n* (-2*I + n)*(2*I + n)*(n + 4*a*x) + (12*(1 - I*a*x)^((I/2)*n)*(-I + a*x)*(I + a*x)*(2 + n^2 + 2*a*n*x + 2*a^2*x^2))/(1 + I*a*x)^((I/2)*n)))/(c*n*(64 + 20*n^2 + n^4)))/(a*c*(36 + n^2)*(c + a^2*c*x^2)^3)
Time = 1.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.87, 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.
| \(\displaystyle \int \frac {e^{n \arctan (a x)}}{\left (a^2 c x^2+c\right )^4} \, dx\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \int \frac {e^{n \arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx}{c \left (n^2+36\right )}+\frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {30 \int \frac {e^{n \arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{c^4 \left (n^2+36\right )}+\frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \left (\frac {12 \int \frac {e^{n \arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{n^2+16}+\frac {(4 a x+n) e^{n \arctan (a x)}}{a \left (n^2+16\right ) \left (a^2 x^2+1\right )^2}\right )}{c^4 \left (n^2+36\right )}+\frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5593 |
| \(\displaystyle \frac {30 \left (\frac {12 \left (\frac {2 \int \frac {e^{n \arctan (a x)}}{a^2 x^2+1}dx}{n^2+4}+\frac {(2 a x+n) e^{n \arctan (a x)}}{a \left (n^2+4\right ) \left (a^2 x^2+1\right )}\right )}{n^2+16}+\frac {(4 a x+n) e^{n \arctan (a x)}}{a \left (n^2+16\right ) \left (a^2 x^2+1\right )^2}\right )}{c^4 \left (n^2+36\right )}+\frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}\) |
| \(\Big \downarrow \) 5594 |
| \(\displaystyle \frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}+\frac {30 \left (\frac {(4 a x+n) e^{n \arctan (a x)}}{a \left (n^2+16\right ) \left (a^2 x^2+1\right )^2}+\frac {12 \left (\frac {(2 a x+n) e^{n \arctan (a x)}}{a \left (n^2+4\right ) \left (a^2 x^2+1\right )}+\frac {2 e^{n \arctan (a x)}}{a n \left (n^2+4\right )}\right )}{n^2+16}\right )}{c^4 \left (n^2+36\right )}\) |
Input:
Int[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]
Output:
(E^(n*ArcTan[a*x])*(n + 6*a*x))/(a*c^4*(36 + n^2)*(1 + a^2*x^2)^3) + (30*( (E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*(16 + n^2)*(1 + a^2*x^2)^2) + (12*((2*E ^(n*ArcTan[a*x]))/(a*n*(4 + n^2)) + (E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*(4 + n^2)*(1 + a^2*x^2))))/(16 + n^2)))/(c^4*(36 + n^2))
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 = 42.05 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92
| method | result | size |
| gosper | \(\frac {\left (720 x^{6} a^{6}+720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} n \,x^{3}+6 a \,n^{5} x +840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}+50 n^{4}+1584 n a x +544 n^{2}+720\right ) {\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} x^{2}+1\right )^{3} c^{4} a n \left (n^{6}+56 n^{4}+784 n^{2}+2304\right )}\) | \(166\) |
| orering | \(\frac {\left (720 x^{6} a^{6}+720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} n \,x^{3}+6 a \,n^{5} x +840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}+50 n^{4}+1584 n a x +544 n^{2}+720\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{n \arctan \left (a x \right )}}{\left (n^{2}+16\right ) \left (n^{4}+40 n^{2}+144\right ) a n \left (a^{2} c \,x^{2}+c \right )^{4}}\) | \(175\) |
| parallelrisch | \(\frac {840 x^{2} {\mathrm e}^{n \arctan \left (a x \right )} a^{2} n^{2}+240 x \,{\mathrm e}^{n \arctan \left (a x \right )} a \,n^{3}+1920 a^{3} {\mathrm e}^{n \arctan \left (a x \right )} x^{3} n +720 a^{5} {\mathrm e}^{n \arctan \left (a x \right )} x^{5} n +1584 x \,{\mathrm e}^{n \arctan \left (a x \right )} n a +{\mathrm e}^{n \arctan \left (a x \right )} n^{6}+50 \,{\mathrm e}^{n \arctan \left (a x \right )} n^{4}+544 \,{\mathrm e}^{n \arctan \left (a x \right )} n^{2}+2160 a^{2} {\mathrm e}^{n \arctan \left (a x \right )} x^{2}+720 \,{\mathrm e}^{n \arctan \left (a x \right )}+2160 a^{4} {\mathrm e}^{n \arctan \left (a x \right )} x^{4}+720 a^{6} {\mathrm e}^{n \arctan \left (a x \right )} x^{6}+360 x^{4} {\mathrm e}^{n \arctan \left (a x \right )} a^{4} n^{2}+120 x^{3} {\mathrm e}^{n \arctan \left (a x \right )} a^{3} n^{3}+30 x^{2} {\mathrm e}^{n \arctan \left (a x \right )} a^{2} n^{4}+6 x \,{\mathrm e}^{n \arctan \left (a x \right )} a \,n^{5}}{c^{4} \left (a^{2} x^{2}+1\right )^{3} \left (n^{2}+16\right ) \left (n^{4}+40 n^{2}+144\right ) a n}\) | \(275\) |
Input:
int(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
(720*a^6*x^6+720*a^5*n*x^5+360*a^4*n^2*x^4+120*a^3*n^3*x^3+2160*a^4*x^4+30 *a^2*n^4*x^2+1920*a^3*n*x^3+6*a*n^5*x+840*a^2*n^2*x^2+n^6+240*a*n^3*x+2160 *a^2*x^2+50*n^4+1584*a*n*x+544*n^2+720)*exp(n*arctan(a*x))/(a^2*x^2+1)^3/c ^4/a/n/(n^6+56*n^4+784*n^2+2304)
Time = 0.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.65 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} + 6 \, a^{4}\right )} x^{4} + 50 \, n^{4} + 120 \, {\left (a^{3} n^{3} + 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} + 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \, {\left (a n^{5} + 40 \, a n^{3} + 264 \, a n\right )} x + 720\right )} e^{\left (n \arctan \left (a x\right )\right )}}{a c^{4} n^{7} + 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} + {\left (a^{7} c^{4} n^{7} + 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} + 2304 \, a^{7} c^{4} n\right )} x^{6} + 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} + 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} + 2304 \, a^{5} c^{4} n\right )} x^{4} + 3 \, {\left (a^{3} c^{4} n^{7} + 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} + 2304 \, a^{3} c^{4} n\right )} x^{2}} \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")
Output:
(720*a^6*x^6 + 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 + 6*a^4)*x^4 + 50*n^4 + 120*(a^3*n^3 + 16*a^3*n)*x^3 + 30*(a^2*n^4 + 28*a^2*n^2 + 72*a^2)*x^2 + 54 4*n^2 + 6*(a*n^5 + 40*a*n^3 + 264*a*n)*x + 720)*e^(n*arctan(a*x))/(a*c^4*n ^7 + 56*a*c^4*n^5 + 784*a*c^4*n^3 + (a^7*c^4*n^7 + 56*a^7*c^4*n^5 + 784*a^ 7*c^4*n^3 + 2304*a^7*c^4*n)*x^6 + 2304*a*c^4*n + 3*(a^5*c^4*n^7 + 56*a^5*c ^4*n^5 + 784*a^5*c^4*n^3 + 2304*a^5*c^4*n)*x^4 + 3*(a^3*c^4*n^7 + 56*a^3*c ^4*n^5 + 784*a^3*c^4*n^3 + 2304*a^3*c^4*n)*x^2)
Timed out. \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(exp(n*atan(a*x))/(a**2*c*x**2+c)**4,x)
Output:
Timed out
\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")
Output:
integrate(e^(n*arctan(a*x))/(a^2*c*x^2 + c)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 1346 vs. \(2 (177) = 354\).
Time = 0.19 (sec) , antiderivative size = 1346, normalized size of antiderivative = 7.44 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(exp(n*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")
Output:
(n^6*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^12 - 6*n^6*e^(n*arctan(a*x))*t an(1/2*arctan(a*x))^10 - 12*n^5*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^11 + 50*n^4*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^12 + 15*n^6*e^(n*arctan(a* x))*tan(1/2*arctan(a*x))^8 + 60*n^5*e^(n*arctan(a*x))*tan(1/2*arctan(a*x)) ^9 - 180*n^4*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^10 - 480*n^3*e^(n*arct an(a*x))*tan(1/2*arctan(a*x))^11 + 544*n^2*e^(n*arctan(a*x))*tan(1/2*arcta n(a*x))^12 - 20*n^6*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^6 - 120*n^5*e^( n*arctan(a*x))*tan(1/2*arctan(a*x))^7 + 270*n^4*e^(n*arctan(a*x))*tan(1/2* arctan(a*x))^8 + 1440*n^3*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^9 + 96*n^ 2*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^10 - 3168*n*e^(n*arctan(a*x))*tan (1/2*arctan(a*x))^11 + 720*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^12 + 15* n^6*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^4 + 120*n^5*e^(n*arctan(a*x))*t an(1/2*arctan(a*x))^5 - 280*n^4*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^6 - 1920*n^3*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^7 + 480*n^2*e^(n*arctan(a *x))*tan(1/2*arctan(a*x))^8 + 480*n*e^(n*arctan(a*x))*tan(1/2*arctan(a*x)) ^9 + 4320*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^10 - 6*n^6*e^(n*arctan(a* x))*tan(1/2*arctan(a*x))^2 - 60*n^5*e^(n*arctan(a*x))*tan(1/2*arctan(a*x)) ^3 + 270*n^4*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^4 + 1920*n^3*e^(n*arct an(a*x))*tan(1/2*arctan(a*x))^5 - 2240*n^2*e^(n*arctan(a*x))*tan(1/2*arcta n(a*x))^6 - 8640*n*e^(n*arctan(a*x))*tan(1/2*arctan(a*x))^7 + 10800*e^(...
Time = 23.24 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.55 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {720\,x^5}{a^2\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {n^6+50\,n^4+544\,n^2+720}{a^7\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {6\,x\,\left (n^4+40\,n^2+264\right )}{a^6\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {120\,x^3\,\left (n^2+16\right )}{a^4\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {360\,x^4\,\left (n^2+6\right )}{a^3\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {30\,x^2\,\left (n^4+28\,n^2+72\right )}{a^5\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}\right )}{\frac {1}{a^6}+x^6+\frac {3\,x^4}{a^2}+\frac {3\,x^2}{a^4}} \] Input:
int(exp(n*atan(a*x))/(c + a^2*c*x^2)^4,x)
Output:
(exp(n*atan(a*x))*((720*x^5)/(a^2*c^4*(784*n^2 + 56*n^4 + n^6 + 2304)) + ( 544*n^2 + 50*n^4 + n^6 + 720)/(a^7*c^4*n*(784*n^2 + 56*n^4 + n^6 + 2304)) + (720*x^6)/(a*c^4*n*(784*n^2 + 56*n^4 + n^6 + 2304)) + (6*x*(40*n^2 + n^4 + 264))/(a^6*c^4*(784*n^2 + 56*n^4 + n^6 + 2304)) + (120*x^3*(n^2 + 16))/ (a^4*c^4*(784*n^2 + 56*n^4 + n^6 + 2304)) + (360*x^4*(n^2 + 6))/(a^3*c^4*n *(784*n^2 + 56*n^4 + n^6 + 2304)) + (30*x^2*(28*n^2 + n^4 + 72))/(a^5*c^4* n*(784*n^2 + 56*n^4 + n^6 + 2304))))/(1/a^6 + x^6 + (3*x^4)/a^2 + (3*x^2)/ a^4)
Time = 0.17 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.53 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {e^{\mathit {atan} \left (a x \right ) n} \left (720 a^{6} x^{6}+720 a^{5} n \,x^{5}+360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} n \,x^{3}+6 a \,n^{5} x +840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}+50 n^{4}+1584 a n x +544 n^{2}+720\right )}{a \,c^{4} n \left (a^{6} n^{6} x^{6}+56 a^{6} n^{4} x^{6}+784 a^{6} n^{2} x^{6}+3 a^{4} n^{6} x^{4}+2304 a^{6} x^{6}+168 a^{4} n^{4} x^{4}+2352 a^{4} n^{2} x^{4}+3 a^{2} n^{6} x^{2}+6912 a^{4} x^{4}+168 a^{2} n^{4} x^{2}+2352 a^{2} n^{2} x^{2}+n^{6}+6912 a^{2} x^{2}+56 n^{4}+784 n^{2}+2304\right )} \] Input:
int(exp(n*atan(a*x))/(a^2*c*x^2+c)^4,x)
Output:
(e**(atan(a*x)*n)*(720*a**6*x**6 + 720*a**5*n*x**5 + 360*a**4*n**2*x**4 + 2160*a**4*x**4 + 120*a**3*n**3*x**3 + 1920*a**3*n*x**3 + 30*a**2*n**4*x**2 + 840*a**2*n**2*x**2 + 2160*a**2*x**2 + 6*a*n**5*x + 240*a*n**3*x + 1584* a*n*x + n**6 + 50*n**4 + 544*n**2 + 720))/(a*c**4*n*(a**6*n**6*x**6 + 56*a **6*n**4*x**6 + 784*a**6*n**2*x**6 + 2304*a**6*x**6 + 3*a**4*n**6*x**4 + 1 68*a**4*n**4*x**4 + 2352*a**4*n**2*x**4 + 6912*a**4*x**4 + 3*a**2*n**6*x** 2 + 168*a**2*n**4*x**2 + 2352*a**2*n**2*x**2 + 6912*a**2*x**2 + n**6 + 56* n**4 + 784*n**2 + 2304))