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

Optimal. Leaf size=40 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2))

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Rubi [A]  time = 0.0499701, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + c*x^4)),x]

[Out]

-1/(2*a*x^2) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2))

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Rubi in Sympy [A]  time = 8.76636, size = 36, normalized size = 0.9 \[ - \frac{1}{2 a x^{2}} - \frac{\sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*x**2) - sqrt(c)*atan(sqrt(c)*x**2/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.0602915, size = 79, normalized size = 1.98 \[ \frac{\sqrt{c} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt{c} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\sqrt{a}}{2 a^{3/2} x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-Sqrt[a] + Sqrt[c]*x^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[c]*x^2*Ar
cTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*a^(3/2)*x^2)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[ -{\frac{c}{2\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{1}{2\,a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^4+a),x)

[Out]

-1/2/a*c/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))-1/2/a/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23313, size = 1, normalized size = 0.02 \[ \left [\frac{x^{2} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2}{4 \, a x^{2}}, \frac{x^{2} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) - 1}{2 \, a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(x^2*sqrt(-c/a)*log((c*x^4 - 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) - 2)/(a*x
^2), 1/2*(x^2*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) - 1)/(a*x^2)]

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Sympy [A]  time = 1.51676, size = 71, normalized size = 1.78 \[ \frac{\sqrt{- \frac{c}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac{\sqrt{- \frac{c}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac{1}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224536, size = 42, normalized size = 1.05 \[ -\frac{c \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a) - 1/2/(a*x^2)