Optimal. Leaf size=33 \[ -3 \sqrt [3]{1+x}+\frac {3}{2} (1+x)^{2/3}+3 \log \left (1+\sqrt [3]{1+x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 196, 45}
\begin {gather*} \frac {3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt [3]{1+x}} \, dx &=\text {Subst}\left (\int \frac {1}{1+\sqrt [3]{x}} \, dx,x,1+x\right )\\ &=3 \text {Subst}\left (\int \frac {x^2}{1+x} \, dx,x,\sqrt [3]{1+x}\right )\\ &=3 \text {Subst}\left (\int \left (-1+x+\frac {1}{1+x}\right ) \, dx,x,\sqrt [3]{1+x}\right )\\ &=-3 \sqrt [3]{1+x}+\frac {3}{2} (1+x)^{2/3}+3 \log \left (1+\sqrt [3]{1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{1+x} \left (-2+\sqrt [3]{1+x}\right )+3 \log \left (1+\sqrt [3]{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 47, normalized size = 1.42
method | result | size |
derivativedivides | \(-3 \left (1+x \right )^{\frac {1}{3}}+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+3 \ln \left (1+\left (1+x \right )^{\frac {1}{3}}\right )\) | \(26\) |
trager | \(-3 \left (1+x \right )^{\frac {1}{3}}+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+\ln \left (-3 \left (1+x \right )^{\frac {2}{3}}-3 \left (1+x \right )^{\frac {1}{3}}-x -2\right )\) | \(36\) |
default | \(\ln \left (2+x \right )+\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2}+2 \ln \left (1+\left (1+x \right )^{\frac {1}{3}}\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}-\left (1+x \right )^{\frac {1}{3}}+1\right )-3 \left (1+x \right )^{\frac {1}{3}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.52, size = 25, normalized size = 0.76 \begin {gather*} \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 25, normalized size = 0.76 \begin {gather*} \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 29, normalized size = 0.88 \begin {gather*} \frac {3 \left (x + 1\right )^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log {\left (\sqrt [3]{x + 1} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 25, normalized size = 0.76 \begin {gather*} \frac {3}{2} \, {\left (x + 1\right )}^{\frac {2}{3}} - 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 25, normalized size = 0.76 \begin {gather*} 3\,\ln \left ({\left (x+1\right )}^{1/3}+1\right )-3\,{\left (x+1\right )}^{1/3}+\frac {3\,{\left (x+1\right )}^{2/3}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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