3.207 \(\int \frac{x^{13}}{\left (a^4+x^4\right )^5} \, dx\)

Optimal. Leaf size=83 \[ -\frac{x^{10}}{16 \left (a^4+x^4\right )^4}-\frac{5 x^6}{96 \left (a^4+x^4\right )^3}+\frac{5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac{5 x^2}{128 \left (a^4+x^4\right )^2}+\frac{5 \tan ^{-1}\left (\frac{x^2}{a^2}\right )}{256 a^6} \]

[Out]

-x^10/(16*(a^4 + x^4)^4) - (5*x^6)/(96*(a^4 + x^4)^3) - (5*x^2)/(128*(a^4 + x^4)
^2) + (5*x^2)/(256*a^4*(a^4 + x^4)) + (5*ArcTan[x^2/a^2])/(256*a^6)

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Rubi [A]  time = 0.0869423, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{x^{10}}{16 \left (a^4+x^4\right )^4}-\frac{5 x^6}{96 \left (a^4+x^4\right )^3}+\frac{5 x^2}{256 a^4 \left (a^4+x^4\right )}-\frac{5 x^2}{128 \left (a^4+x^4\right )^2}+\frac{5 \tan ^{-1}\left (\frac{x^2}{a^2}\right )}{256 a^6} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a^4 + x^4)^5,x]

[Out]

-x^10/(16*(a^4 + x^4)^4) - (5*x^6)/(96*(a^4 + x^4)^3) - (5*x^2)/(128*(a^4 + x^4)
^2) + (5*x^2)/(256*a^4*(a^4 + x^4)) + (5*ArcTan[x^2/a^2])/(256*a^6)

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Rubi in Sympy [A]  time = 5.98969, size = 75, normalized size = 0.9 \[ - \frac{x^{10}}{16 \left (a^{4} + x^{4}\right )^{4}} - \frac{5 x^{6}}{96 \left (a^{4} + x^{4}\right )^{3}} - \frac{5 x^{2}}{128 \left (a^{4} + x^{4}\right )^{2}} + \frac{5 x^{2}}{256 a^{4} \left (a^{4} + x^{4}\right )} + \frac{5 \operatorname{atan}{\left (\frac{x^{2}}{a^{2}} \right )}}{256 a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(a**4+x**4)**5,x)

[Out]

-x**10/(16*(a**4 + x**4)**4) - 5*x**6/(96*(a**4 + x**4)**3) - 5*x**2/(128*(a**4
+ x**4)**2) + 5*x**2/(256*a**4*(a**4 + x**4)) + 5*atan(x**2/a**2)/(256*a**6)

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Mathematica [A]  time = 0.0351716, size = 62, normalized size = 0.75 \[ \frac{15 \tan ^{-1}\left (\frac{x^2}{a^2}\right )-\frac{a^2 x^2 \left (15 a^{12}+55 a^8 x^4+73 a^4 x^8-15 x^{12}\right )}{\left (a^4+x^4\right )^4}}{768 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a^4 + x^4)^5,x]

[Out]

(-((a^2*x^2*(15*a^12 + 55*a^8*x^4 + 73*a^4*x^8 - 15*x^12))/(a^4 + x^4)^4) + 15*A
rcTan[x^2/a^2])/(768*a^6)

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Maple [A]  time = 0.017, size = 56, normalized size = 0.7 \[{\frac{1}{2\, \left ({a}^{4}+{x}^{4} \right ) ^{4}} \left ({\frac{5\,{x}^{14}}{128\,{a}^{4}}}-{\frac{73\,{x}^{10}}{384}}-{\frac{55\,{x}^{6}{a}^{4}}{384}}-{\frac{5\,{a}^{8}{x}^{2}}{128}} \right ) }+{\frac{5}{256\,{a}^{6}}\arctan \left ({\frac{{x}^{2}}{{a}^{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(a^4+x^4)^5,x)

[Out]

1/2*(5/128/a^4*x^14-73/384*x^10-55/384*x^6*a^4-5/128*a^8*x^2)/(a^4+x^4)^4+5/256*
arctan(x^2/a^2)/a^6

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Maxima [A]  time = 1.51007, size = 112, normalized size = 1.35 \[ -\frac{15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \,{\left (a^{20} + 4 \, a^{16} x^{4} + 6 \, a^{12} x^{8} + 4 \, a^{8} x^{12} + a^{4} x^{16}\right )}} + \frac{5 \, \arctan \left (\frac{x^{2}}{a^{2}}\right )}{256 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(a^4 + x^4)^5,x, algorithm="maxima")

[Out]

-1/768*(15*a^12*x^2 + 55*a^8*x^6 + 73*a^4*x^10 - 15*x^14)/(a^20 + 4*a^16*x^4 + 6
*a^12*x^8 + 4*a^8*x^12 + a^4*x^16) + 5/256*arctan(x^2/a^2)/a^6

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Fricas [A]  time = 0.200782, size = 153, normalized size = 1.84 \[ -\frac{15 \, a^{14} x^{2} + 55 \, a^{10} x^{6} + 73 \, a^{6} x^{10} - 15 \, a^{2} x^{14} - 15 \,{\left (a^{16} + 4 \, a^{12} x^{4} + 6 \, a^{8} x^{8} + 4 \, a^{4} x^{12} + x^{16}\right )} \arctan \left (\frac{x^{2}}{a^{2}}\right )}{768 \,{\left (a^{22} + 4 \, a^{18} x^{4} + 6 \, a^{14} x^{8} + 4 \, a^{10} x^{12} + a^{6} x^{16}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(a^4 + x^4)^5,x, algorithm="fricas")

[Out]

-1/768*(15*a^14*x^2 + 55*a^10*x^6 + 73*a^6*x^10 - 15*a^2*x^14 - 15*(a^16 + 4*a^1
2*x^4 + 6*a^8*x^8 + 4*a^4*x^12 + x^16)*arctan(x^2/a^2))/(a^22 + 4*a^18*x^4 + 6*a
^14*x^8 + 4*a^10*x^12 + a^6*x^16)

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Sympy [A]  time = 92.0704, size = 102, normalized size = 1.23 \[ \frac{- 15 a^{12} x^{2} - 55 a^{8} x^{6} - 73 a^{4} x^{10} + 15 x^{14}}{768 a^{20} + 3072 a^{16} x^{4} + 4608 a^{12} x^{8} + 3072 a^{8} x^{12} + 768 a^{4} x^{16}} + \frac{- \frac{5 i \log{\left (- i a^{2} + x^{2} \right )}}{512} + \frac{5 i \log{\left (i a^{2} + x^{2} \right )}}{512}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(a**4+x**4)**5,x)

[Out]

(-15*a**12*x**2 - 55*a**8*x**6 - 73*a**4*x**10 + 15*x**14)/(768*a**20 + 3072*a**
16*x**4 + 4608*a**12*x**8 + 3072*a**8*x**12 + 768*a**4*x**16) + (-5*I*log(-I*a**
2 + x**2)/512 + 5*I*log(I*a**2 + x**2)/512)/a**6

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GIAC/XCAS [A]  time = 0.203804, size = 78, normalized size = 0.94 \[ \frac{5 \, \arctan \left (\frac{x^{2}}{a^{2}}\right )}{256 \, a^{6}} - \frac{15 \, a^{12} x^{2} + 55 \, a^{8} x^{6} + 73 \, a^{4} x^{10} - 15 \, x^{14}}{768 \,{\left (a^{4} + x^{4}\right )}^{4} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(a^4 + x^4)^5,x, algorithm="giac")

[Out]

5/256*arctan(x^2/a^2)/a^6 - 1/768*(15*a^12*x^2 + 55*a^8*x^6 + 73*a^4*x^10 - 15*x
^14)/((a^4 + x^4)^4*a^4)