3.676 \(\int \frac{1}{x \left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=54 \[ -\frac{\log \left (a+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{1}{4 a^2 \left (a+c x^4\right )}+\frac{1}{8 a \left (a+c x^4\right )^2} \]

[Out]

1/(8*a*(a + c*x^4)^2) + 1/(4*a^2*(a + c*x^4)) + Log[x]/a^3 - Log[a + c*x^4]/(4*a
^3)

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Rubi [A]  time = 0.079981, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{1}{4 a^2 \left (a+c x^4\right )}+\frac{1}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + c*x^4)^3),x]

[Out]

1/(8*a*(a + c*x^4)^2) + 1/(4*a^2*(a + c*x^4)) + Log[x]/a^3 - Log[a + c*x^4]/(4*a
^3)

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Rubi in Sympy [A]  time = 10.8501, size = 49, normalized size = 0.91 \[ \frac{1}{8 a \left (a + c x^{4}\right )^{2}} + \frac{1}{4 a^{2} \left (a + c x^{4}\right )} + \frac{\log{\left (x^{4} \right )}}{4 a^{3}} - \frac{\log{\left (a + c x^{4} \right )}}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(c*x**4+a)**3,x)

[Out]

1/(8*a*(a + c*x**4)**2) + 1/(4*a**2*(a + c*x**4)) + log(x**4)/(4*a**3) - log(a +
c*x**4)/(4*a**3)

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Mathematica [A]  time = 0.0624152, size = 43, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+2 c x^4\right )}{\left (a+c x^4\right )^2}-2 \log \left (a+c x^4\right )+8 \log (x)}{8 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + c*x^4)^3),x]

[Out]

((a*(3*a + 2*c*x^4))/(a + c*x^4)^2 + 8*Log[x] - 2*Log[a + c*x^4])/(8*a^3)

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Maple [A]  time = 0.022, size = 49, normalized size = 0.9 \[{\frac{1}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{1}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+a)^3,x)

[Out]

1/8/a/(c*x^4+a)^2+1/4/a^2/(c*x^4+a)+ln(x)/a^3-1/4*ln(c*x^4+a)/a^3

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Maxima [A]  time = 1.42709, size = 81, normalized size = 1.5 \[ \frac{2 \, c x^{4} + 3 \, a}{8 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} - \frac{\log \left (c x^{4} + a\right )}{4 \, a^{3}} + \frac{\log \left (x^{4}\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x),x, algorithm="maxima")

[Out]

1/8*(2*c*x^4 + 3*a)/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) - 1/4*log(c*x^4 + a)/a^3 +
1/4*log(x^4)/a^3

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Fricas [A]  time = 0.231196, size = 122, normalized size = 2.26 \[ \frac{2 \, a c x^{4} + 3 \, a^{2} - 2 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right ) + 8 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (x\right )}{8 \,{\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x),x, algorithm="fricas")

[Out]

1/8*(2*a*c*x^4 + 3*a^2 - 2*(c^2*x^8 + 2*a*c*x^4 + a^2)*log(c*x^4 + a) + 8*(c^2*x
^8 + 2*a*c*x^4 + a^2)*log(x))/(a^3*c^2*x^8 + 2*a^4*c*x^4 + a^5)

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Sympy [A]  time = 10.5943, size = 56, normalized size = 1.04 \[ \frac{3 a + 2 c x^{4}}{8 a^{4} + 16 a^{3} c x^{4} + 8 a^{2} c^{2} x^{8}} + \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (\frac{a}{c} + x^{4} \right )}}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**4+a)**3,x)

[Out]

(3*a + 2*c*x**4)/(8*a**4 + 16*a**3*c*x**4 + 8*a**2*c**2*x**8) + log(x)/a**3 - lo
g(a/c + x**4)/(4*a**3)

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GIAC/XCAS [A]  time = 0.223924, size = 80, normalized size = 1.48 \[ \frac{{\rm ln}\left (x^{4}\right )}{4 \, a^{3}} - \frac{{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{3}} + \frac{3 \, c^{2} x^{8} + 8 \, a c x^{4} + 6 \, a^{2}}{8 \,{\left (c x^{4} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x),x, algorithm="giac")

[Out]

1/4*ln(x^4)/a^3 - 1/4*ln(abs(c*x^4 + a))/a^3 + 1/8*(3*c^2*x^8 + 8*a*c*x^4 + 6*a^
2)/((c*x^4 + a)^2*a^3)