Optimal. Leaf size=54 \[ 12 \sqrt [4]{1-2 e^{x/3}}-6 \tan ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2320, 52, 65,
218, 212, 209} \begin {gather*} -6 \text {ArcTan}\left (\sqrt [4]{1-2 e^{x/3}}\right )+12 \sqrt [4]{1-2 e^{x/3}}-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 218
Rule 2320
Rubi steps
\begin {align*} \int \sqrt [4]{1-2 e^{x/3}} \, dx &=3 \text {Subst}\left (\int \frac {\sqrt [4]{1-2 x}}{x} \, dx,x,e^{x/3}\right )\\ &=12 \sqrt [4]{1-2 e^{x/3}}+3 \text {Subst}\left (\int \frac {1}{(1-2 x)^{3/4} x} \, dx,x,e^{x/3}\right )\\ &=12 \sqrt [4]{1-2 e^{x/3}}-6 \text {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {x^4}{2}} \, dx,x,\sqrt [4]{1-2 e^{x/3}}\right )\\ &=12 \sqrt [4]{1-2 e^{x/3}}-6 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1-2 e^{x/3}}\right )\\ &=12 \sqrt [4]{1-2 e^{x/3}}-6 \tan ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 1.00 \begin {gather*} 12 \sqrt [4]{1-2 e^{x/3}}-6 \tan ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 57, normalized size = 1.06
method | result | size |
derivativedivides | \(12 \left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+1\right )-6 \arctan \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}\right )\) | \(57\) |
default | \(12 \left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+1\right )-6 \arctan \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}\right )\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.59, size = 56, normalized size = 1.04 \begin {gather*} 12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 56, normalized size = 1.04 \begin {gather*} 12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [4]{1 - 2 e^{\frac {x}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 57, normalized size = 1.06 \begin {gather*} 12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left | {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 33, normalized size = 0.61 \begin {gather*} \frac {12\,{\left (2-4\,{\mathrm {e}}^{x/3}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {{\mathrm {e}}^{-\frac {x}{3}}}{2}\right )}{{\left (2-{\mathrm {e}}^{-\frac {x}{3}}\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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