3.18 \(\int \frac{\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac{3}{2} (\sin (x)-\cos (x))^{2/3} \]

[Out]

(3*(-Cos[x] + Sin[x])^(2/3))/2

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Rubi [A]  time = 0.0430681, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{3}{2} (\sin (x)-\cos (x))^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(Cos[x] + Sin[x])/(-Cos[x] + Sin[x])^(1/3),x]

[Out]

(3*(-Cos[x] + Sin[x])^(2/3))/2

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Rubi in Sympy [A]  time = 2.8627, size = 12, normalized size = 0.8 \[ \frac{3 \left (\sin{\left (x \right )} - \cos{\left (x \right )}\right )^{\frac{2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((cos(x)+sin(x))/(-cos(x)+sin(x))**(1/3),x)

[Out]

3*(sin(x) - cos(x))**(2/3)/2

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Mathematica [A]  time = 0.0289661, size = 15, normalized size = 1. \[ \frac{3}{2} (\sin (x)-\cos (x))^{2/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Cos[x] + Sin[x])/(-Cos[x] + Sin[x])^(1/3),x]

[Out]

(3*(-Cos[x] + Sin[x])^(2/3))/2

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Maple [A]  time = 0.017, size = 12, normalized size = 0.8 \[{\frac{3}{2} \left ( -\cos \left ( x \right ) +\sin \left ( x \right ) \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((cos(x)+sin(x))/(-cos(x)+sin(x))^(1/3),x)

[Out]

3/2*(-cos(x)+sin(x))^(2/3)

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Maxima [A]  time = 1.34672, size = 15, normalized size = 1. \[ \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x))/(-cos(x) + sin(x))^(1/3),x, algorithm="maxima")

[Out]

3/2*(-cos(x) + sin(x))^(2/3)

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Fricas [A]  time = 0.223118, size = 15, normalized size = 1. \[ \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x))/(-cos(x) + sin(x))^(1/3),x, algorithm="fricas")

[Out]

3/2*(-cos(x) + sin(x))^(2/3)

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Sympy [A]  time = 0.482085, size = 12, normalized size = 0.8 \[ \frac{3 \left (\sin{\left (x \right )} - \cos{\left (x \right )}\right )^{\frac{2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x)+sin(x))/(-cos(x)+sin(x))**(1/3),x)

[Out]

3*(sin(x) - cos(x))**(2/3)/2

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GIAC/XCAS [A]  time = 0.22928, size = 15, normalized size = 1. \[ \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x))/(-cos(x) + sin(x))^(1/3),x, algorithm="giac")

[Out]

3/2*(-cos(x) + sin(x))^(2/3)