3.296 \(\int \frac{\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx\)

Optimal. Leaf size=48 \[ 4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0483133, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ 4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^(1/3))^(3/4)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.30034, size = 42, normalized size = 0.88 \[ 4 \left (- 2 \sqrt [3]{x} + 1\right )^{\frac{3}{4}} + 6 \operatorname{atan}{\left (\sqrt [4]{- 2 \sqrt [3]{x} + 1} \right )} - 6 \operatorname{atanh}{\left (\sqrt [4]{- 2 \sqrt [3]{x} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0467015, size = 62, normalized size = 1.29 \[ \frac{-6\ 2^{3/4} \sqrt [4]{2-\frac{1}{\sqrt [3]{x}}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2 \sqrt [3]{x}}\right )-8 \sqrt [3]{x}+4}{\sqrt [4]{1-2 \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(4 - 8*x^(1/3) - 6*2^(3/4)*(2 - x^(-1/3))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4,
1/(2*x^(1/3))])/(1 - 2*x^(1/3))^(1/4)

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Maple [A]  time = 0.013, size = 53, normalized size = 1.1 \[ 4\, \left ( 1-2\,\sqrt [3]{x} \right ) ^{3/4}+3\,\ln \left ( -1+\sqrt [4]{1-2\,\sqrt [3]{x}} \right ) -3\,\ln \left ( 1+\sqrt [4]{1-2\,\sqrt [3]{x}} \right ) +6\,\arctan \left ( \sqrt [4]{1-2\,\sqrt [3]{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x^(1/3))^(3/4)/x,x)

[Out]

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

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Maxima [A]  time = 1.58913, size = 70, normalized size = 1.46 \[ 4 \,{\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{3}{4}} + 6 \, \arctan \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22315, size = 70, normalized size = 1.46 \[ 4 \,{\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{3}{4}} + 6 \, \arctan \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, x^{\frac{1}{3}} + 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.69654, size = 51, normalized size = 1.06 \[ - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt [4]{x} e^{\frac{3 i \pi }{4}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{1}{2 \sqrt [3]{x}}} \right )}}{\Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(3/4)*x**(1/4)*exp(3*I*pi/4)*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), 1/(2*
x**(1/3)))/gamma(1/4)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError