Optimal. Leaf size=47 \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0651268, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(a^(2/3)*C + 2*C*x^2)/(a + 8*x^3),x]
[Out]
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Rubi in Sympy [A] time = 9.77557, size = 44, normalized size = 0.94 \[ \frac{C \log{\left (\sqrt [3]{a} + 2 x \right )}}{4} - \frac{\sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{4 x}{3}\right )}{\sqrt [3]{a}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**(2/3)*C+2*C*x**2)/(8*x**3+a),x)
[Out]
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Mathematica [A] time = 0.0459204, size = 72, normalized size = 1.53 \[ \frac{1}{12} C \left (-\log \left (a^{2/3}-2 \sqrt [3]{a} x+4 x^2\right )+\log \left (a+8 x^3\right )+2 \log \left (\sqrt [3]{a}+2 x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a^(2/3)*C + 2*C*x^2)/(a + 8*x^3),x]
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Maple [B] time = 0.009, size = 84, normalized size = 1.8 \[{\frac{C{8}^{{\frac{2}{3}}}}{24}\ln \left ( x+{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ) }-{\frac{C{8}^{{\frac{2}{3}}}}{48}\ln \left ({x}^{2}-{\frac{x{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ) }+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}-1 \right ) } \right ) }+{\frac{C\ln \left ( 8\,{x}^{3}+a \right ) }{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^(2/3)*C+2*C*x^2)/(8*x^3+a),x)
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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((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232747, size = 61, normalized size = 1.3 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} C \log \left (2 \, a^{\frac{2}{3}} x + a\right ) + 2 \, C \arctan \left (\frac{4 \, \sqrt{3} a^{\frac{2}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.807761, size = 85, normalized size = 1.81 \[ C \left (\frac{\log{\left (\frac{\sqrt [3]{a}}{2} + x \right )}}{4} - \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} - \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12} + \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} + \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**(2/3)*C+2*C*x**2)/(8*x**3+a),x)
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GIAC/XCAS [A] time = 0.243111, size = 147, normalized size = 3.13 \[ \frac{\sqrt{3}{\left (\sqrt{3} a i + a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{12 \, a} - \frac{{\left (\sqrt{3} a i - 3 \, a\right )} C{\rm ln}\left (x^{2} + \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} x + \frac{1}{4} \, \left (-a\right )^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (C \left (-a\right )^{\frac{2}{3}} + 2 \, C a^{\frac{2}{3}}\right )} \left (-a\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} \right |}\right )}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="giac")
[Out]