Optimal. Leaf size=12 \[ x-\frac{1}{3} \cot ^3(x)+\cot (x) \]
[Out]
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Rubi [A] time = 0.0149954, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ x-\frac{1}{3} \cot ^3(x)+\cot (x) \]
Antiderivative was successfully verified.
[In] Int[Cot[x]^4,x]
[Out]
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Rubi in Sympy [A] time = 0.520162, size = 14, normalized size = 1.17 \[ x + \frac{1}{\tan{\left (x \right )}} - \frac{1}{3 \tan ^{3}{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cot(x)**4,x)
[Out]
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Mathematica [A] time = 0.00525028, size = 18, normalized size = 1.5 \[ x+\frac{4 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
[In] Integrate[Cot[x]^4,x]
[Out]
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Maple [A] time = 0.004, size = 14, normalized size = 1.2 \[ -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{3}}{3}}+\cot \left ( x \right ) -{\frac{\pi }{2}}+x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cot(x)^4,x)
[Out]
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Maxima [A] time = 1.55278, size = 22, normalized size = 1.83 \[ x + \frac{3 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cot(x)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246982, size = 65, normalized size = 5.42 \[ \frac{4 \, \cos \left (2 \, x\right )^{2} + 3 \,{\left (x \cos \left (2 \, x\right ) - x\right )} \sin \left (2 \, x\right ) + 2 \, \cos \left (2 \, x\right ) - 2}{3 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cot(x)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.052207, size = 19, normalized size = 1.58 \[ x + \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} - \frac{\cos ^{3}{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cot(x)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.207786, size = 46, normalized size = 3.83 \[ \frac{1}{24} \, \tan \left (\frac{1}{2} \, x\right )^{3} + x + \frac{15 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{24 \, \tan \left (\frac{1}{2} \, x\right )^{3}} - \frac{5}{8} \, \tan \left (\frac{1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cot(x)^4,x, algorithm="giac")
[Out]