3.528 \(\int \sqrt [4]{1-2 e^{x/3}} \, dx\)

Optimal. Leaf size=54 \[ 12 \sqrt [4]{1-2 e^{x/3}}-6 \tan ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \]

[Out]

12*(1 - 2*E^(x/3))^(1/4) - 6*ArcTan[(1 - 2*E^(x/3))^(1/4)] - 6*ArcTanh[(1 - 2*E^
(x/3))^(1/4)]

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Rubi [A]  time = 0.0507944, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ 12 \sqrt [4]{1-2 e^{x/3}}-6 \tan ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*E^(x/3))^(1/4),x]

[Out]

12*(1 - 2*E^(x/3))^(1/4) - 6*ArcTan[(1 - 2*E^(x/3))^(1/4)] - 6*ArcTanh[(1 - 2*E^
(x/3))^(1/4)]

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Rubi in Sympy [A]  time = 2.53313, size = 42, normalized size = 0.78 \[ 12 \sqrt [4]{- 2 e^{\frac{x}{3}} + 1} - 6 \operatorname{atan}{\left (\sqrt [4]{- 2 e^{\frac{x}{3}} + 1} \right )} - 6 \operatorname{atanh}{\left (\sqrt [4]{- 2 e^{\frac{x}{3}} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*exp(1/3*x))**(1/4),x)

[Out]

12*(-2*exp(x/3) + 1)**(1/4) - 6*atan((-2*exp(x/3) + 1)**(1/4)) - 6*atanh((-2*exp
(x/3) + 1)**(1/4))

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Mathematica [C]  time = 0.0568485, size = 70, normalized size = 1.3 \[ -\frac{2 \left (\sqrt [4]{2} \left (2-e^{-x/3}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{e^{-x/3}}{2}\right )+12 e^{x/3}-6\right )}{\left (1-2 e^{x/3}\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*E^(x/3))^(1/4),x]

[Out]

(-2*(-6 + 12*E^(x/3) + 2^(1/4)*(2 - E^(-x/3))^(3/4)*Hypergeometric2F1[3/4, 3/4,
7/4, 1/(2*E^(x/3))]))/(1 - 2*E^(x/3))^(3/4)

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Maple [A]  time = 0.01, size = 57, normalized size = 1.1 \[ 12\,\sqrt [4]{1-2\,{{\rm e}^{x/3}}}+3\,\ln \left ( -1+\sqrt [4]{1-2\,{{\rm e}^{x/3}}} \right ) -3\,\ln \left ( 1+\sqrt [4]{1-2\,{{\rm e}^{x/3}}} \right ) -6\,\arctan \left ( \sqrt [4]{1-2\,{{\rm e}^{x/3}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*exp(1/3*x))^(1/4),x)

[Out]

12*(1-2*exp(1/3*x))^(1/4)+3*ln(-1+(1-2*exp(1/3*x))^(1/4))-3*ln(1+(1-2*exp(1/3*x)
)^(1/4))-6*arctan((1-2*exp(1/3*x))^(1/4))

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Maxima [A]  time = 1.56773, size = 76, normalized size = 1.41 \[ 12 \,{\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*e^(1/3*x) + 1)^(1/4),x, algorithm="maxima")

[Out]

12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*log((-2*e^(
1/3*x) + 1)^(1/4) + 1) + 3*log((-2*e^(1/3*x) + 1)^(1/4) - 1)

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Fricas [A]  time = 0.223122, size = 76, normalized size = 1.41 \[ 12 \,{\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*e^(1/3*x) + 1)^(1/4),x, algorithm="fricas")

[Out]

12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*log((-2*e^(
1/3*x) + 1)^(1/4) + 1) + 3*log((-2*e^(1/3*x) + 1)^(1/4) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt [4]{- 2 e^{\frac{x}{3}} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*exp(1/3*x))**(1/4),x)

[Out]

Integral((-2*exp(x/3) + 1)**(1/4), x)

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GIAC/XCAS [A]  time = 0.209881, size = 77, normalized size = 1.43 \[ 12 \,{\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}}\right ) - 3 \,{\rm ln}\left ({\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} + 1\right ) + 3 \,{\rm ln}\left ({\left |{\left (-2 \, e^{\left (\frac{1}{3} \, x\right )} + 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*e^(1/3*x) + 1)^(1/4),x, algorithm="giac")

[Out]

12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*ln((-2*e^(1
/3*x) + 1)^(1/4) + 1) + 3*ln(abs((-2*e^(1/3*x) + 1)^(1/4) - 1))