3.213 \(\int \frac{1}{-\sqrt{1+x}+(1+x)^{2/3}} \, dx\)

Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]

[Out]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[1 - (1 + x)^(1/6)]

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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]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[1 - (1 + x)^(1/6)]

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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]

6*(x + 1)**(1/6) + 6*log(-(x + 1)**(1/6) + 1) + 6*Integral(x, (x, (x + 1)**(1/6)
))

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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]

3*(2*(1 + x)^(1/6) + (1 + x)^(1/3) + 2*Log[1 - (1 + x)^(1/6)])

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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]

6*(1+x)^(1/6)+3*(1+x)^(1/3)+ln(x)+2*ln(-1+(1+x)^(1/6))-ln((1+x)^(1/3)+(1+x)^(1/6
)+1)-2*ln(1+(1+x)^(1/6))+ln((1+x)^(1/3)-(1+x)^(1/6)+1)-ln(1+(1+x)^(1/2))+ln(-1+(
1+x)^(1/2))+2*ln(-1+(1+x)^(1/3))-ln((1+x)^(2/3)+(1+x)^(1/3)+1)

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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]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*log((x + 1)^(1/6) - 1)

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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]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*log((x + 1)^(1/6) - 1)

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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]

Integral(1/((x + 1)**(2/3) - sqrt(x + 1)), x)

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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]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*ln(abs((x + 1)^(1/6) - 1))