3.132 \(\int \frac{1}{x^4 \left (a^4-x^4\right )} \, dx\)

Optimal. Leaf size=37 \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7}-\frac{1}{3 a^4 x^3} \]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) + ArcTanh[x/a]/(2*a^7)

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Rubi [A]  time = 0.0321356, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7}-\frac{1}{3 a^4 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) + ArcTanh[x/a]/(2*a^7)

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Rubi in Sympy [A]  time = 3.58675, size = 29, normalized size = 0.78 \[ - \frac{1}{3 a^{4} x^{3}} + \frac{\operatorname{atan}{\left (\frac{x}{a} \right )}}{2 a^{7}} + \frac{\operatorname{atanh}{\left (\frac{x}{a} \right )}}{2 a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(a**4-x**4),x)

[Out]

-1/(3*a**4*x**3) + atan(x/a)/(2*a**7) + atanh(x/a)/(2*a**7)

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Mathematica [A]  time = 0.010457, size = 48, normalized size = 1.3 \[ -\frac{\log (a-x)}{4 a^7}+\frac{\log (a+x)}{4 a^7}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}-\frac{1}{3 a^4 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) - Log[a - x]/(4*a^7) + Log[a + x]/(4*a^7)

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Maple [A]  time = 0.013, size = 41, normalized size = 1.1 \[{\frac{\ln \left ( a+x \right ) }{4\,{a}^{7}}}-{\frac{1}{3\,{a}^{4}{x}^{3}}}+{\frac{1}{2\,{a}^{7}}\arctan \left ({\frac{x}{a}} \right ) }-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a^4-x^4),x)

[Out]

1/4*ln(a+x)/a^7-1/3/a^4/x^3+1/2*arctan(x/a)/a^7-1/4/a^7*ln(-a+x)

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Maxima [A]  time = 1.51284, size = 54, normalized size = 1.46 \[ \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{\log \left (a + x\right )}{4 \, a^{7}} - \frac{\log \left (-a + x\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^4),x, algorithm="maxima")

[Out]

1/2*arctan(x/a)/a^7 + 1/4*log(a + x)/a^7 - 1/4*log(-a + x)/a^7 - 1/3/(a^4*x^3)

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Fricas [A]  time = 0.296172, size = 61, normalized size = 1.65 \[ \frac{6 \, x^{3} \arctan \left (\frac{x}{a}\right ) + 3 \, x^{3} \log \left (a + x\right ) - 3 \, x^{3} \log \left (-a + x\right ) - 4 \, a^{3}}{12 \, a^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^4),x, algorithm="fricas")

[Out]

1/12*(6*x^3*arctan(x/a) + 3*x^3*log(a + x) - 3*x^3*log(-a + x) - 4*a^3)/(a^7*x^3
)

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Sympy [A]  time = 0.738934, size = 48, normalized size = 1.3 \[ - \frac{1}{3 a^{4} x^{3}} - \frac{\frac{\log{\left (- a + x \right )}}{4} - \frac{\log{\left (a + x \right )}}{4} + \frac{i \log{\left (- i a + x \right )}}{4} - \frac{i \log{\left (i a + x \right )}}{4}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(a**4-x**4),x)

[Out]

-1/(3*a**4*x**3) - (log(-a + x)/4 - log(a + x)/4 + I*log(-I*a + x)/4 - I*log(I*a
+ x)/4)/a**7

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GIAC/XCAS [A]  time = 0.202227, size = 57, normalized size = 1.54 \[ \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{4 \, a^{7}} - \frac{{\rm ln}\left ({\left | -a + x \right |}\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^4),x, algorithm="giac")

[Out]

1/2*arctan(x/a)/a^7 + 1/4*ln(abs(a + x))/a^7 - 1/4*ln(abs(-a + x))/a^7 - 1/3/(a^
4*x^3)