Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(i-5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt {c+a^2 c x^2}} \] Output:
1/120*(I-5*a*x)*(a^2*x^2+1)^(1/2)/a^3/c^13/(1-I*a*x)^10/(1+I*a*x)^15/(a^2* c*x^2+c)^(1/2)
Time = 0.61 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(1+5 i a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (-i+a x)^{15} (i+a x)^{10} \sqrt {c+a^2 c x^2}} \] Input:
Integrate[x^2/(E^((5*I)*ArcTan[a*x])*(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)^15*(I + a*x)^ 10*Sqrt[c + a^2*c*x^2])
Time = 0.74 (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 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {x^2}{(1-i a x)^{11} (i a x+1)^{16}}dx}{c^{13} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 91 |
\(\displaystyle \frac {(-5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt {a^2 c x^2+c}}\) |
Input:
Int[x^2/(E^((5*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(27/2)),x]
Output:
((I - 5*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(1 - I*a*x)^10*(1 + I*a*x)^1 5*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 = 0.36 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {\frac {i x}{24 a^{2}}+\frac {1}{120 a^{3}}}{c^{13} \left (a^{2} x^{2}+1\right )^{\frac {19}{2}} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x -i\right )^{5}}\) | \(50\) |
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 | \(-\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (-5 a x +i\right ) \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{120 a^{3} \left (i a x +1\right )^{5} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(58\) |
orering | \(-\frac {x^{3} \left (-a^{22} x^{22}+5 i a^{21} x^{21}+40 i a^{19} x^{19}+50 a^{18} x^{18}+126 i a^{17} x^{17}+280 a^{16} x^{16}+160 i a^{15} x^{15}+765 a^{14} x^{14}-105 i a^{13} x^{13}+1248 a^{12} x^{12}-720 i a^{11} x^{11}+1260 a^{10} x^{10}-1260 i a^{9} x^{9}+720 a^{8} x^{8}-1248 i a^{7} x^{7}+105 x^{6} a^{6}-765 i x^{5} a^{5}-160 a^{4} x^{4}-280 i x^{3} a^{3}-126 a^{2} x^{2}-50 i a x -40\right ) \left (a^{2} x^{2}+1\right )^{\frac {7}{2}}}{120 \left (i a x +1\right )^{5} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(215\) |
Input:
int(x^2/(1+I*a*x)^5*(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(27/2),x,method=_RETUR NVERBOSE)
Output:
1/c^13/(a^2*x^2+1)^(19/2)/(c*(a^2*x^2+1))^(1/2)*(1/24*I/a^2*x+1/120/a^3)/( a*x-I)^5
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(x^2/(1+I*a*x)^5*(a^2*x^2+1)^(5/2)/(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^ 17*x^20 + 280*I*a^16*x^19 - 160*a^15*x^18 + 765*I*a^14*x^17 + 105*a^13*x^1 6 + 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)*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^2 1*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)
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(x**2/(1+I*a*x)**5*(a**2*x**2+1)**(5/2)/(a**2*c*x**2+c)**(27/2),x )
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (53) = 106\).
Time = 0.12 (sec) , antiderivative size = 274, normalized size of antiderivative = 4.22 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {5 i \, a \sqrt {c} x + \sqrt {c}}{120 \, {\left (a^{28} c^{14} x^{25} - 5 i \, a^{27} c^{14} x^{24} - 40 i \, a^{25} c^{14} x^{22} - 50 \, a^{24} c^{14} x^{21} - 126 i \, a^{23} c^{14} x^{20} - 280 \, a^{22} c^{14} x^{19} - 160 i \, a^{21} c^{14} x^{18} - 765 \, a^{20} c^{14} x^{17} + 105 i \, a^{19} c^{14} x^{16} - 1248 \, a^{18} c^{14} x^{15} + 720 i \, a^{17} c^{14} x^{14} - 1260 \, a^{16} c^{14} x^{13} + 1260 i \, a^{15} c^{14} x^{12} - 720 \, a^{14} c^{14} x^{11} + 1248 i \, a^{13} c^{14} x^{10} - 105 \, a^{12} c^{14} x^{9} + 765 i \, a^{11} c^{14} x^{8} + 160 \, a^{10} c^{14} x^{7} + 280 i \, a^{9} c^{14} x^{6} + 126 \, a^{8} c^{14} x^{5} + 50 i \, a^{7} c^{14} x^{4} + 40 \, a^{6} c^{14} x^{3} + 5 \, a^{4} c^{14} x - i \, a^{3} c^{14}\right )}} \] Input:
integrate(x^2/(1+I*a*x)^5*(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(27/2),x, algori thm="maxima")
Output:
1/120*(5*I*a*sqrt(c)*x + sqrt(c))/(a^28*c^14*x^25 - 5*I*a^27*c^14*x^24 - 4 0*I*a^25*c^14*x^22 - 50*a^24*c^14*x^21 - 126*I*a^23*c^14*x^20 - 280*a^22*c ^14*x^19 - 160*I*a^21*c^14*x^18 - 765*a^20*c^14*x^17 + 105*I*a^19*c^14*x^1 6 - 1248*a^18*c^14*x^15 + 720*I*a^17*c^14*x^14 - 1260*a^16*c^14*x^13 + 126 0*I*a^15*c^14*x^12 - 720*a^14*c^14*x^11 + 1248*I*a^13*c^14*x^10 - 105*a^12 *c^14*x^9 + 765*I*a^11*c^14*x^8 + 160*a^10*c^14*x^7 + 280*I*a^9*c^14*x^6 + 126*a^8*c^14*x^5 + 50*I*a^7*c^14*x^4 + 40*a^6*c^14*x^3 + 5*a^4*c^14*x - I *a^3*c^14)
Exception generated. \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/(1+I*a*x)^5*(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(27/2),x, algori thm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo r the root of a polynomial with parameters. This might be wrong.The choice was done
Time = 27.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {c^2\,\sqrt {a^2\,x^2+1}\,{\left (a\,x+1{}\mathrm {i}\right )}^5\,\left (1+a\,x\,5{}\mathrm {i}\right )}{120\,a^3\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{31/2}} \] Input:
int((x^2*(a^2*x^2 + 1)^(5/2))/((c + a^2*c*x^2)^(27/2)*(a*x*1i + 1)^5),x)
Output:
(c^2*(a^2*x^2 + 1)^(1/2)*(a*x + 1i)^5*(a*x*5i + 1))/(120*a^3*(c*(a^2*x^2 + 1))^(31/2))
\[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {x^{2}}{a^{27} i \,x^{27}+5 a^{26} x^{26}+a^{25} i \,x^{25}+45 a^{24} x^{24}-50 a^{23} i \,x^{23}+166 a^{22} x^{22}-330 a^{21} i \,x^{21}+286 a^{20} x^{20}-1045 a^{19} i \,x^{19}+55 a^{18} x^{18}-2013 a^{17} i \,x^{17}-825 a^{16} x^{16}-2508 a^{15} i \,x^{15}-1980 a^{14} x^{14}-1980 a^{13} i \,x^{13}-2508 a^{12} x^{12}-825 a^{11} i \,x^{11}-2013 a^{10} x^{10}+55 a^{9} i \,x^{9}-1045 a^{8} x^{8}+286 a^{7} i \,x^{7}-330 a^{6} x^{6}+166 a^{5} i \,x^{5}-50 a^{4} x^{4}+45 a^{3} i \,x^{3}+a^{2} x^{2}+5 a i x +1}d x \right )}{c^{14}} \] Input:
int(x^2/(1+I*a*x)^5*(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(27/2),x)
Output:
(sqrt(c)*int(x**2/(a**27*i*x**27 + 5*a**26*x**26 + a**25*i*x**25 + 45*a**2 4*x**24 - 50*a**23*i*x**23 + 166*a**22*x**22 - 330*a**21*i*x**21 + 286*a** 20*x**20 - 1045*a**19*i*x**19 + 55*a**18*x**18 - 2013*a**17*i*x**17 - 825* a**16*x**16 - 2508*a**15*i*x**15 - 1980*a**14*x**14 - 1980*a**13*i*x**13 - 2508*a**12*x**12 - 825*a**11*i*x**11 - 2013*a**10*x**10 + 55*a**9*i*x**9 - 1045*a**8*x**8 + 286*a**7*i*x**7 - 330*a**6*x**6 + 166*a**5*i*x**5 - 50* a**4*x**4 + 45*a**3*i*x**3 + a**2*x**2 + 5*a*i*x + 1),x))/c**14