3.4.79 \(\int \frac {e^{-4 i \text {ArcTan}(a x)} x^2}{(c+a^2 c x^2)^9} \, dx\) [379]

Optimal. Leaf size=38 \[ \frac {i-4 a x}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \]

[Out]

1/60*(I-4*a*x)/a^3/c^9/(1-I*a*x)^6/(1+I*a*x)^10

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Rubi [A]
time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5190, 82} \begin {gather*} \frac {-4 a x+i}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(E^((4*I)*ArcTan[a*x])*(c + a^2*c*x^2)^9),x]

[Out]

(I - 4*a*x)/(60*a^3*c^9*(1 - I*a*x)^6*(1 + I*a*x)^10)

Rule 82

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rule 5190

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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] && (Int
egerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {e^{-4 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx &=\frac {\int \frac {x^2}{(1-i a x)^7 (1+i a x)^{11}} \, dx}{c^9}\\ &=\frac {i-4 a x}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 36, normalized size = 0.95 \begin {gather*} \frac {i-4 a x}{60 a^3 c^9 (-i+a x)^{10} (i+a x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(E^((4*I)*ArcTan[a*x])*(c + a^2*c*x^2)^9),x]

[Out]

(I - 4*a*x)/(60*a^3*c^9*(-I + a*x)^10*(I + a*x)^6)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (33 ) = 66\).
time = 0.32, size = 218, normalized size = 5.74

method	result	size
risch	\(\frac {\frac {i}{60 a^{3}}-\frac {x}{15 a^{2}}}{c^{9} \left (a x -i\right )^{10} \left (a x +i\right )^{6}}\)	\(34\)
gosper	\(-\frac {\left (-4 a x +i\right ) \left (a x +i\right ) \left (-a x +i\right )}{60 \left (a^{2} x^{2}+1\right )^{7} c^{9} \left (i a x +1\right )^{4} a^{3}}\)	\(49\)
norman	\(\frac {-\frac {i a \,x^{4}}{c}+\frac {x^{3}}{3 c}-\frac {a^{2} x^{5}}{15 c}-\frac {2 i a^{3} x^{6}}{c}-\frac {7 i a^{5} x^{8}}{2 c}-\frac {21 i a^{7} x^{10}}{5 c}-\frac {7 i a^{9} x^{12}}{2 c}-\frac {2 i a^{11} x^{14}}{c}-\frac {3 i a^{13} x^{16}}{4 c}-\frac {i a^{15} x^{18}}{6 c}-\frac {i a^{17} x^{20}}{60 c}}{\left (a^{2} x^{2}+1\right )^{10} c^{8}}\)	\(142\)
default	\(\frac {\frac {21 i}{8192 a^{3} \left (-a x +i\right )^{4}}+\frac {i}{1280 a^{3} \left (-a x +i\right )^{10}}-\frac {i}{1024 a^{3} \left (-a x +i\right )^{8}}-\frac {7 i}{6144 a^{3} \left (-a x +i\right )^{6}}-\frac {165 i}{65536 a^{3} \left (-a x +i\right )^{2}}+\frac {1}{768 a^{3} \left (-a x +i\right )^{9}}-\frac {21}{10240 a^{3} \left (-a x +i\right )^{5}}+\frac {11}{4096 a^{3} \left (-a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (-a x +i\right )}+\frac {13 i}{16384 a^{3} \left (a x +i\right )^{4}}-\frac {i}{12288 a^{3} \left (a x +i\right )^{6}}-\frac {121 i}{65536 a^{3} \left (a x +i\right )^{2}}-\frac {7}{20480 a^{3} \left (a x +i\right )^{5}}+\frac {11}{8192 a^{3} \left (a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (a x +i\right )}}{c^{9}}\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+I*a*x)^4*(a^2*x^2+1)^2/(a^2*c*x^2+c)^9,x,method=_RETURNVERBOSE)

[Out]

1/c^9*(21/8192*I/a^3/(I-a*x)^4+1/1280*I/a^3/(I-a*x)^10-1/1024*I/a^3/(I-a*x)^8-7/6144*I/a^3/(I-a*x)^6-165/65536
*I/a^3/(I-a*x)^2+1/768/a^3/(I-a*x)^9-21/10240/a^3/(I-a*x)^5+11/4096/a^3/(I-a*x)^3-143/65536/a^3/(I-a*x)+13/163
84*I/a^3/(I+a*x)^4-1/12288*I/a^3/(I+a*x)^6-121/65536*I/a^3/(I+a*x)^2-7/20480/a^3/(I+a*x)^5+11/8192/a^3/(I+a*x)
^3-143/65536/a^3/(I+a*x))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^4*(a^2*x^2+1)^2/(a^2*c*x^2+c)^9,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (30) = 60\).
time = 2.67, size = 169, normalized size = 4.45 \begin {gather*} -\frac {4 \, a x - i}{60 \, {\left (a^{19} c^{9} x^{16} - 4 i \, a^{18} c^{9} x^{15} - 20 i \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} - 36 i \, a^{14} c^{9} x^{11} - 64 \, a^{13} c^{9} x^{10} - 20 i \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} + 20 i \, a^{10} c^{9} x^{7} - 64 \, a^{9} c^{9} x^{6} + 36 i \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} + 20 i \, a^{6} c^{9} x^{3} + 4 i \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^4*(a^2*x^2+1)^2/(a^2*c*x^2+c)^9,x, algorithm="fricas")

[Out]

-1/60*(4*a*x - I)/(a^19*c^9*x^16 - 4*I*a^18*c^9*x^15 - 20*I*a^16*c^9*x^13 - 20*a^15*c^9*x^12 - 36*I*a^14*c^9*x
^11 - 64*a^13*c^9*x^10 - 20*I*a^12*c^9*x^9 - 90*a^11*c^9*x^8 + 20*I*a^10*c^9*x^7 - 64*a^9*c^9*x^6 + 36*I*a^8*c
^9*x^5 - 20*a^7*c^9*x^4 + 20*I*a^6*c^9*x^3 + 4*I*a^4*c^9*x + a^3*c^9)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (31) = 62\).
time = 0.85, size = 192, normalized size = 5.05 \begin {gather*} \frac {- 4 a x + i}{60 a^{19} c^{9} x^{16} - 240 i a^{18} c^{9} x^{15} - 1200 i a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} - 2160 i a^{14} c^{9} x^{11} - 3840 a^{13} c^{9} x^{10} - 1200 i a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} + 1200 i a^{10} c^{9} x^{7} - 3840 a^{9} c^{9} x^{6} + 2160 i a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} + 1200 i a^{6} c^{9} x^{3} + 240 i a^{4} c^{9} x + 60 a^{3} c^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(1+I*a*x)**4*(a**2*x**2+1)**2/(a**2*c*x**2+c)**9,x)

[Out]

(-4*a*x + I)/(60*a**19*c**9*x**16 - 240*I*a**18*c**9*x**15 - 1200*I*a**16*c**9*x**13 - 1200*a**15*c**9*x**12 -
2160*I*a**14*c**9*x**11 - 3840*a**13*c**9*x**10 - 1200*I*a**12*c**9*x**9 - 5400*a**11*c**9*x**8 + 1200*I*a**1
0*c**9*x**7 - 3840*a**9*c**9*x**6 + 2160*I*a**8*c**9*x**5 - 1200*a**7*c**9*x**4 + 1200*I*a**6*c**9*x**3 + 240*
I*a**4*c**9*x + 60*a**3*c**9)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (30) = 60\).
time = 0.40, size = 139, normalized size = 3.66 \begin {gather*} -\frac {2145 \, a^{5} x^{5} + 12540 i \, a^{4} x^{4} - 30030 \, a^{3} x^{3} - 37080 i \, a^{2} x^{2} + 23841 \, a x + 6476 i}{983040 \, {\left (a x + i\right )}^{6} a^{3} c^{9}} + \frac {2145 \, a^{9} x^{9} - 21780 i \, a^{8} x^{8} - 99660 \, a^{7} x^{7} + 270480 i \, a^{6} x^{6} + 481446 \, a^{5} x^{5} - 584920 i \, a^{4} x^{4} - 486220 \, a^{3} x^{3} + 265680 i \, a^{2} x^{2} + 84065 \, a x - 9908 i}{983040 \, {\left (a x - i\right )}^{10} a^{3} c^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^4*(a^2*x^2+1)^2/(a^2*c*x^2+c)^9,x, algorithm="giac")

[Out]

-1/983040*(2145*a^5*x^5 + 12540*I*a^4*x^4 - 30030*a^3*x^3 - 37080*I*a^2*x^2 + 23841*a*x + 6476*I)/((a*x + I)^6
*a^3*c^9) + 1/983040*(2145*a^9*x^9 - 21780*I*a^8*x^8 - 99660*a^7*x^7 + 270480*I*a^6*x^6 + 481446*a^5*x^5 - 584
920*I*a^4*x^4 - 486220*a^3*x^3 + 265680*I*a^2*x^2 + 84065*a*x - 9908*I)/((a*x - I)^10*a^3*c^9)

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Mupad [B]
time = 3.40, size = 159, normalized size = 4.18 \begin {gather*} \frac {-4\,a^5\,x^5-a^4\,x^4\,15{}\mathrm {i}+20\,a^3\,x^3+a^2\,x^2\,10{}\mathrm {i}+1{}\mathrm {i}}{60\,a^{23}\,c^9\,x^{20}+600\,a^{21}\,c^9\,x^{18}+2700\,a^{19}\,c^9\,x^{16}+7200\,a^{17}\,c^9\,x^{14}+12600\,a^{15}\,c^9\,x^{12}+15120\,a^{13}\,c^9\,x^{10}+12600\,a^{11}\,c^9\,x^8+7200\,a^9\,c^9\,x^6+2700\,a^7\,c^9\,x^4+600\,a^5\,c^9\,x^2+60\,a^3\,c^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a^2*x^2 + 1)^2)/((c + a^2*c*x^2)^9*(a*x*1i + 1)^4),x)

[Out]

(a^2*x^2*10i + 20*a^3*x^3 - a^4*x^4*15i - 4*a^5*x^5 + 1i)/(60*a^3*c^9 + 600*a^5*c^9*x^2 + 2700*a^7*c^9*x^4 + 7
200*a^9*c^9*x^6 + 12600*a^11*c^9*x^8 + 15120*a^13*c^9*x^10 + 12600*a^15*c^9*x^12 + 7200*a^17*c^9*x^14 + 2700*a
^19*c^9*x^16 + 600*a^21*c^9*x^18 + 60*a^23*c^9*x^20)

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