Integrand size = 15, antiderivative size = 54 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1-2 e^{x/3}}-6 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2320, 52, 65, 218, 212, 209} \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=-6 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right )+12 \sqrt [4]{1-2 e^{x/3}} \]
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 218
Rule 2320
Rubi steps \begin{align*} \text {integral}& = 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 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1-2 e^{x/3}}-6 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 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 | \(12 \left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+1\right )-6 \arctan \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}\right )\) | \(57\) |
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Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.54 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1 - 2 e^{\frac {x}{3}}} + 3 \log {\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} - 1 \right )} - 3 \log {\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} + 1 \right )} - 6 \operatorname {atan}{\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left | {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=\frac {12\,{\left (2-4\,{\mathrm {e}}^{x/3}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {{\mathrm {e}}^{-\frac {x}{3}}}{2}\right )}{{\left (2-{\mathrm {e}}^{-\frac {x}{3}}\right )}^{1/4}} \]
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