Optimal. Leaf size=124 \[ \frac{\log \left (\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a+1}-\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} (a+1)^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.13763, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{\log \left (\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a+1}-\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} (a+1)^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Int[(1 + a - b*x^3)^(-1),x]
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Rubi in Sympy [A] time = 25.1931, size = 117, normalized size = 0.94 \[ - \frac{\log{\left (\sqrt [3]{b} x - \sqrt [3]{a + 1} \right )}}{3 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} + \frac{\log{\left (b^{\frac{2}{3}} x^{2} + \sqrt [3]{b} x \sqrt [3]{a + 1} + \left (a + 1\right )^{\frac{2}{3}} \right )}}{6 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + 1}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x**3+a+1),x)
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Mathematica [A] time = 0.114059, size = 124, normalized size = 1. \[ \frac{(-1)^{2/3} \left (\log \left (-\sqrt [3]{-1} \sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+(-1)^{2/3} b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a+1}+\sqrt [3]{-1} \sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a+1}}-1}{\sqrt{3}}\right )\right )}{6 (a+1)^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + a - b*x^3)^(-1),x]
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Maple [A] time = 0.007, size = 106, normalized size = 0.9 \[ -{\frac{1}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{1+a}{b}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{1+a}{b}}}+ \left ({\frac{1+a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{1+a}{b}}}}}}+1 \right ) } \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x^3+a+1),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(-1/(b*x^3 - a - 1),x, algorithm="maxima")
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Fricas [A] time = 0.218992, size = 176, normalized size = 1.42 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{2}{3}} x^{2} - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}}{\left (a + 1\right )} x + a^{2} + 2 \, a + 1\right ) - 2 \, \sqrt{3} \log \left (\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}} x + a + 1\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}} x - \sqrt{3}{\left (a + 1\right )}}{3 \,{\left (a + 1\right )}}\right )\right )}}{18 \, \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b*x^3 - a - 1),x, algorithm="fricas")
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Sympy [A] time = 0.652328, size = 34, normalized size = 0.27 \[ - \operatorname{RootSum}{\left (t^{3} \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log{\left (- 3 t a - 3 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x**3+a+1),x)
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GIAC/XCAS [A] time = 0.240769, size = 177, normalized size = 1.43 \[ \frac{{\left (a b^{2} + b^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a + 1}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a + 1}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b + \sqrt{3} b} + \frac{{\left (a b^{2} + b^{2}\right )}^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (\frac{a + 1}{b}\right )^{\frac{1}{3}} + \left (\frac{a + 1}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b + b\right )}} - \frac{\left (\frac{a + 1}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a + 1}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b*x^3 - a - 1),x, algorithm="giac")
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