Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=-\frac {(i+5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {c+a^2 c x^2}} \] Output:
-1/120*(I+5*a*x)*(a^2*x^2+1)^(1/2)/a^3/c^13/(1-I*a*x)^15/(1+I*a*x)^10/(a^2 *c*x^2+c)^(1/2)
Time = 0.59 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(1-5 i a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (-i+a x)^{10} (i+a x)^{15} \sqrt {c+a^2 c x^2}} \] Input:
Integrate[(E^((5*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(27/2),x]
Output:
((1 - (5*I)*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(-I + a*x)^10*(I + a*x)^ 15*Sqrt[c + a^2*c*x^2])
Time = 0.72 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5608, 5605, 91}
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.
\(\displaystyle \int \frac {x^2 e^{5 i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{27/2}} \, dx\) |
\(\Big \downarrow \) 5608 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{5 i \arctan (a x)} x^2}{\left (a^2 x^2+1\right )^{27/2}}dx}{c^{13} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5605 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {x^2}{(1-i a x)^{16} (i a x+1)^{11}}dx}{c^{13} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 91 |
\(\displaystyle -\frac {(5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {a^2 c x^2+c}}\) |
Input:
Int[(E^((5*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(27/2),x]
Output:
-1/120*((I + 5*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^13*(1 - I*a*x)^15*(1 + I*a*x )^10*Sqrt[c + a^2*c*x^2])
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3 ) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2 , 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*( b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a* d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I* (n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Integer Q[p] || GtQ[c, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart [p]) Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (5 i a x -1\right )}{120 \sqrt {a^{2} x^{2}+1}\, c^{14} a^{3} \left (a x +i\right )^{15} \left (-a x +i\right )^{10}}\) | \(57\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (5 a x +i\right ) \left (i a x +1\right )^{5}}{120 a^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(58\) |
risch | \(\frac {\sqrt {a^{2} x^{2}+1}\, \left (\frac {1}{120 a^{3}}-\frac {i x}{24 a^{2}}\right )}{c^{13} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{15} \left (a x -i\right )^{10}}\) | \(58\) |
orering | \(\frac {x^{3} \left (a^{22} x^{22}+5 i a^{21} x^{21}+40 i a^{19} x^{19}-50 a^{18} x^{18}+126 i a^{17} x^{17}-280 a^{16} x^{16}+160 i a^{15} x^{15}-765 a^{14} x^{14}-105 i a^{13} x^{13}-1248 a^{12} x^{12}-720 i a^{11} x^{11}-1260 a^{10} x^{10}-1260 i a^{9} x^{9}-720 a^{8} x^{8}-1248 i a^{7} x^{7}-105 x^{6} a^{6}-765 i x^{5} a^{5}+160 a^{4} x^{4}-280 i x^{3} a^{3}+126 a^{2} x^{2}-50 i a x +40\right ) \left (i a x +1\right )^{5}}{120 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(214\) |
Input:
int((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x,method=_RETUR NVERBOSE)
Output:
-1/120/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)*(5*I*a*x-1)/c^14/a^3/(I+a*x )^15/(I-a*x)^10
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (53) = 106\).
Time = 0.19 (sec) , antiderivative size = 496, normalized size of antiderivative = 7.63 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {{\left (i \, a^{22} x^{25} - 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 i \, a^{18} x^{21} - 126 \, a^{17} x^{20} - 280 i \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 i \, a^{14} x^{17} + 105 \, a^{13} x^{16} - 1248 i \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 i \, a^{10} x^{13} + 1260 \, a^{9} x^{12} - 720 i \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 i \, a^{6} x^{9} + 765 \, a^{5} x^{8} + 160 i \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 i \, a^{2} x^{5} + 50 \, a x^{4} + 40 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} + 5 i \, a^{26} c^{14} x^{26} + a^{25} c^{14} x^{25} + 45 i \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 i \, a^{22} c^{14} x^{22} - 330 \, a^{21} c^{14} x^{21} + 286 i \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 i \, a^{18} c^{14} x^{18} - 2013 \, a^{17} c^{14} x^{17} - 825 i \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 i \, a^{14} c^{14} x^{14} - 1980 \, a^{13} c^{14} x^{13} - 2508 i \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 i \, a^{10} c^{14} x^{10} + 55 \, a^{9} c^{14} x^{9} - 1045 i \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 i \, a^{6} c^{14} x^{6} + 166 \, a^{5} c^{14} x^{5} - 50 i \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + i \, a^{2} c^{14} x^{2} + 5 \, a c^{14} x + i \, c^{14}\right )}} \] Input:
integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algori thm="fricas")
Output:
1/120*(I*a^22*x^25 - 5*a^21*x^24 - 40*a^19*x^22 - 50*I*a^18*x^21 - 126*a^1 7*x^20 - 280*I*a^16*x^19 - 160*a^15*x^18 - 765*I*a^14*x^17 + 105*a^13*x^16 - 1248*I*a^12*x^15 + 720*a^11*x^14 - 1260*I*a^10*x^13 + 1260*a^9*x^12 - 7 20*I*a^8*x^11 + 1248*a^7*x^10 - 105*I*a^6*x^9 + 765*a^5*x^8 + 160*I*a^4*x^ 7 + 280*a^3*x^6 + 126*I*a^2*x^5 + 50*a*x^4 + 40*I*x^3)*sqrt(a^2*c*x^2 + c) *sqrt(a^2*x^2 + 1)/(a^27*c^14*x^27 + 5*I*a^26*c^14*x^26 + a^25*c^14*x^25 + 45*I*a^24*c^14*x^24 - 50*a^23*c^14*x^23 + 166*I*a^22*c^14*x^22 - 330*a^21 *c^14*x^21 + 286*I*a^20*c^14*x^20 - 1045*a^19*c^14*x^19 + 55*I*a^18*c^14*x ^18 - 2013*a^17*c^14*x^17 - 825*I*a^16*c^14*x^16 - 2508*a^15*c^14*x^15 - 1 980*I*a^14*c^14*x^14 - 1980*a^13*c^14*x^13 - 2508*I*a^12*c^14*x^12 - 825*a ^11*c^14*x^11 - 2013*I*a^10*c^14*x^10 + 55*a^9*c^14*x^9 - 1045*I*a^8*c^14* x^8 + 286*a^7*c^14*x^7 - 330*I*a^6*c^14*x^6 + 166*a^5*c^14*x^5 - 50*I*a^4* c^14*x^4 + 45*a^3*c^14*x^3 + I*a^2*c^14*x^2 + 5*a*c^14*x + I*c^14)
Timed out. \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Timed out} \] Input:
integrate((1+I*a*x)**5/(a**2*x**2+1)**(5/2)*x**2/(a**2*c*x**2+c)**(27/2),x )
Output:
Timed out
Exception generated. \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algori thm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algori thm="giac")
Output:
integrate((I*a*x + 1)^5*x^2/((a^2*c*x^2 + c)^(27/2)*(a^2*x^2 + 1)^(5/2)), x)
Time = 2.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=-\frac {c\,{\left (a\,x-\mathrm {i}\right )}^5\,\left (5\,a\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{120\,a^3\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{29/2}\,\sqrt {a^2\,x^2+1}} \] Input:
int((x^2*(a*x*1i + 1)^5)/((c + a^2*c*x^2)^(27/2)*(a^2*x^2 + 1)^(5/2)),x)
Output:
-(c*(a*x - 1i)^5*(5*a*x + 1i)*1i)/(120*a^3*(c*(a^2*x^2 + 1))^(29/2)*(a^2*x ^2 + 1)^(1/2))
Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.77 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {\sqrt {c}\, \left (-5 a^{6} i \,x^{6}-24 a^{5} x^{5}+45 a^{4} i \,x^{4}+40 a^{3} x^{3}-15 a^{2} i \,x^{2}-i \right )}{120 a^{3} c^{14} \left (a^{30} x^{30}+15 a^{28} x^{28}+105 a^{26} x^{26}+455 a^{24} x^{24}+1365 a^{22} x^{22}+3003 a^{20} x^{20}+5005 a^{18} x^{18}+6435 a^{16} x^{16}+6435 a^{14} x^{14}+5005 a^{12} x^{12}+3003 a^{10} x^{10}+1365 a^{8} x^{8}+455 a^{6} x^{6}+105 a^{4} x^{4}+15 a^{2} x^{2}+1\right )} \] Input:
int((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x)
Output:
(sqrt(c)*( - 5*a**6*i*x**6 - 24*a**5*x**5 + 45*a**4*i*x**4 + 40*a**3*x**3 - 15*a**2*i*x**2 - i))/(120*a**3*c**14*(a**30*x**30 + 15*a**28*x**28 + 105 *a**26*x**26 + 455*a**24*x**24 + 1365*a**22*x**22 + 3003*a**20*x**20 + 500 5*a**18*x**18 + 6435*a**16*x**16 + 6435*a**14*x**14 + 5005*a**12*x**12 + 3 003*a**10*x**10 + 1365*a**8*x**8 + 455*a**6*x**6 + 105*a**4*x**4 + 15*a**2 *x**2 + 1))