Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
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Rubi [A] time = 0.0377612, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[(-Sqrt[1 + x] + (1 + x)^(2/3))^(-1),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 6 \sqrt [6]{x + 1} + 6 \log{\left (- \sqrt [6]{x + 1} + 1 \right )} + 6 \int ^{\sqrt [6]{x + 1}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((1+x)**(2/3)-(1+x)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0169437, size = 33, normalized size = 1. \[ 3 \left (\sqrt [3]{x+1}+2 \sqrt [6]{x+1}+2 \log \left (1-\sqrt [6]{x+1}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(-Sqrt[1 + x] + (1 + x)^(2/3))^(-1),x]
[Out]
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Maple [B] time = 0.03, size = 111, normalized size = 3.4 \[ 6\,\sqrt [6]{1+x}+3\,\sqrt [3]{1+x}+\ln \left ( x \right ) +2\,\ln \left ( -1+\sqrt [6]{1+x} \right ) -\ln \left ( \sqrt [3]{1+x}+\sqrt [6]{1+x}+1 \right ) -2\,\ln \left ( 1+\sqrt [6]{1+x} \right ) +\ln \left ( \sqrt [3]{1+x}-\sqrt [6]{1+x}+1 \right ) -\ln \left ( 1+\sqrt{1+x} \right ) +\ln \left ( -1+\sqrt{1+x} \right ) +2\,\ln \left ( -1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}+\sqrt [3]{1+x}+1 \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((1+x)^(2/3)-(1+x)^(1/2)),x)
[Out]
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Maxima [A] time = 1.38007, size = 34, normalized size = 1.03 \[ 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac{1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(2/3) - sqrt(x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202291, size = 34, normalized size = 1.03 \[ 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac{1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(2/3) - sqrt(x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 1\right )^{\frac{2}{3}} - \sqrt{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((1+x)**(2/3)-(1+x)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.209296, size = 35, normalized size = 1.06 \[ 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 6 \,{\left (x + 1\right )}^{\frac{1}{6}} + 6 \,{\rm ln}\left ({\left |{\left (x + 1\right )}^{\frac{1}{6}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(2/3) - sqrt(x + 1)),x, algorithm="giac")
[Out]