3.180 \(\int \frac{-16+x^{3/4}}{16+x^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ x-128 \sqrt [4]{x}+\frac{256}{3} \sqrt [3]{2} \log \left (\sqrt [4]{x}+2 \sqrt [3]{2}\right )-\frac{128}{3} \sqrt [3]{2} \log \left (\sqrt{x}-2 \sqrt [3]{2} \sqrt [4]{x}+4\ 2^{2/3}\right )-\frac{256 \sqrt [3]{2} \tan ^{-1}\left (\frac{\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-128*x^(1/4) + x - (256*2^(1/3)*ArcTan[(2^(1/3) - x^(1/4))/(2^(1/3)*Sqrt[3])])/S
qrt[3] + (256*2^(1/3)*Log[2*2^(1/3) + x^(1/4)])/3 - (128*2^(1/3)*Log[4*2^(2/3) -
2*2^(1/3)*x^(1/4) + Sqrt[x]])/3

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Rubi [A]  time = 0.203454, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ x-128 \sqrt [4]{x}+\frac{256}{3} \sqrt [3]{2} \log \left (\sqrt [4]{x}+2 \sqrt [3]{2}\right )-\frac{128}{3} \sqrt [3]{2} \log \left (\sqrt{x}-2 \sqrt [3]{2} \sqrt [4]{x}+4\ 2^{2/3}\right )-\frac{256 \sqrt [3]{2} \tan ^{-1}\left (\frac{\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-16 + x^(3/4))/(16 + x^(3/4)),x]

[Out]

-128*x^(1/4) + x - (256*2^(1/3)*ArcTan[(2^(1/3) - x^(1/4))/(2^(1/3)*Sqrt[3])])/S
qrt[3] + (256*2^(1/3)*Log[2*2^(1/3) + x^(1/4)])/3 - (128*2^(1/3)*Log[4*2^(2/3) -
2*2^(1/3)*x^(1/4) + Sqrt[x]])/3

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Rubi in Sympy [A]  time = 19.8801, size = 99, normalized size = 0.95 \[ - 128 \sqrt [4]{x} + x + \frac{256 \sqrt [3]{2} \log{\left (\sqrt [4]{x} + 2 \sqrt [3]{2} \right )}}{3} - \frac{128 \sqrt [3]{2} \log{\left (- 2 \sqrt [3]{2} \sqrt [4]{x} + \sqrt{x} + 4 \cdot 2^{\frac{2}{3}} \right )}}{3} - \frac{256 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2^{\frac{2}{3}} \sqrt [4]{x}}{6} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-16+x**(3/4))/(16+x**(3/4)),x)

[Out]

-128*x**(1/4) + x + 256*2**(1/3)*log(x**(1/4) + 2*2**(1/3))/3 - 128*2**(1/3)*log
(-2*2**(1/3)*x**(1/4) + sqrt(x) + 4*2**(2/3))/3 - 256*2**(1/3)*sqrt(3)*atan(sqrt
(3)*(-2**(2/3)*x**(1/4)/6 + 1/3))/3

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Mathematica [A]  time = 0.0496975, size = 102, normalized size = 0.98 \[ x-128 \sqrt [4]{x}+\frac{256}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [4]{x}+4\right )-\frac{128}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt{x}-2\ 2^{2/3} \sqrt [4]{x}+8\right )+\frac{256 \sqrt [3]{2} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [4]{x}-2}{2 \sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-16 + x^(3/4))/(16 + x^(3/4)),x]

[Out]

-128*x^(1/4) + x + (256*2^(1/3)*ArcTan[(-2 + 2^(2/3)*x^(1/4))/(2*Sqrt[3])])/Sqrt
[3] + (256*2^(1/3)*Log[4 + 2^(2/3)*x^(1/4)])/3 - (128*2^(1/3)*Log[8 - 2*2^(2/3)*
x^(1/4) + 2^(1/3)*Sqrt[x]])/3

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Maple [A]  time = 0.009, size = 66, normalized size = 0.6 \[ x-128\,\sqrt [4]{x}+{\frac{128\,\sqrt [3]{16}}{3}\ln \left ( \sqrt [4]{x}+\sqrt [3]{16} \right ) }-{\frac{64\,\sqrt [3]{16}}{3}\ln \left ( \sqrt{x}-\sqrt [4]{x}\sqrt [3]{16}+{16}^{{\frac{2}{3}}} \right ) }+{\frac{128\,\sqrt [3]{16}\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{16}^{{\frac{2}{3}}}}{8}\sqrt [4]{x}}-1 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-16+x^(3/4))/(16+x^(3/4)),x)

[Out]

x-128*x^(1/4)+128/3*16^(1/3)*ln(x^(1/4)+16^(1/3))-64/3*16^(1/3)*ln(x^(1/2)-x^(1/
4)*16^(1/3)+16^(2/3))+128/3*16^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(1/8*16^(2/3)*x^
(1/4)-1))

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Maxima [A]  time = 1.64799, size = 90, normalized size = 0.87 \[ \frac{256}{3} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (-\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} - x^{\frac{1}{4}}\right )}\right ) - \frac{128}{3} \cdot 2^{\frac{1}{3}} \log \left (\left (4 \cdot 2^{\frac{2}{3}}\right ) - 2 \cdot 2^{\frac{1}{3}} x^{\frac{1}{4}} + \sqrt{x}\right ) + \frac{256}{3} \cdot 2^{\frac{1}{3}} \log \left (\left (2 \cdot 2^{\frac{1}{3}}\right ) + x^{\frac{1}{4}}\right ) + x - 128 \, x^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(3/4) - 16)/(x^(3/4) + 16),x, algorithm="maxima")

[Out]

256/3*sqrt(3)*2^(1/3)*arctan(-1/6*sqrt(3)*2^(2/3)*(2^(1/3) - x^(1/4))) - 128/3*2
^(1/3)*log((4*2^(2/3)) - 2*2^(1/3)*x^(1/4) + sqrt(x)) + 256/3*2^(1/3)*log((2*2^(
1/3)) + x^(1/4)) + x - 128*x^(1/4)

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Fricas [A]  time = 0.242223, size = 123, normalized size = 1.18 \[ -\frac{1}{9} \, \sqrt{3}{\left (128 \, \sqrt{3} 2^{\frac{1}{3}} \log \left (4 \cdot 2^{\frac{2}{3}} - 2 \cdot 2^{\frac{1}{3}} x^{\frac{1}{4}} + \sqrt{x}\right ) - 256 \, \sqrt{3} 2^{\frac{1}{3}} \log \left (2 \cdot 2^{\frac{1}{3}} + x^{\frac{1}{4}}\right ) - 3 \, \sqrt{3} x - 768 \cdot 2^{\frac{1}{3}} \arctan \left (-\frac{1}{6} \cdot 2^{\frac{2}{3}}{\left (\sqrt{3} 2^{\frac{1}{3}} - \sqrt{3} x^{\frac{1}{4}}\right )}\right ) + 384 \, \sqrt{3} x^{\frac{1}{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(3/4) - 16)/(x^(3/4) + 16),x, algorithm="fricas")

[Out]

-1/9*sqrt(3)*(128*sqrt(3)*2^(1/3)*log(4*2^(2/3) - 2*2^(1/3)*x^(1/4) + sqrt(x)) -
256*sqrt(3)*2^(1/3)*log(2*2^(1/3) + x^(1/4)) - 3*sqrt(3)*x - 768*2^(1/3)*arctan
(-1/6*2^(2/3)*(sqrt(3)*2^(1/3) - sqrt(3)*x^(1/4))) + 384*sqrt(3)*x^(1/4))

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Sympy [A]  time = 20.2673, size = 102, normalized size = 0.98 \[ - 128 \sqrt [4]{x} + x + \frac{256 \sqrt [3]{2} \log{\left (\sqrt [4]{x} + 2 \sqrt [3]{2} \right )}}{3} - \frac{128 \sqrt [3]{2} \log{\left (- 2 \sqrt [3]{2} \sqrt [4]{x} + \sqrt{x} + 4 \cdot 2^{\frac{2}{3}} \right )}}{3} + \frac{256 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \sqrt [4]{x}}{6} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-16+x**(3/4))/(16+x**(3/4)),x)

[Out]

-128*x**(1/4) + x + 256*2**(1/3)*log(x**(1/4) + 2*2**(1/3))/3 - 128*2**(1/3)*log
(-2*2**(1/3)*x**(1/4) + sqrt(x) + 4*2**(2/3))/3 + 256*2**(1/3)*sqrt(3)*atan(2**(
2/3)*sqrt(3)*x**(1/4)/6 - sqrt(3)/3)/3

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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((x^(3/4) - 16)/(x^(3/4) + 16),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError