Optimal. Leaf size=20 \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]
[Out]
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Rubi [A] time = 0.024199, antiderivative size = 20, 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 \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]
Antiderivative was successfully verified.
[In] Int[Cot[x]^5,x]
[Out]
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Rubi in Sympy [A] time = 0.529923, size = 20, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} \right )} + \frac{1}{2 \tan ^{2}{\left (x \right )}} - \frac{1}{4 \tan ^{4}{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/tan(x)**5,x)
[Out]
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Mathematica [A] time = 0.00550819, size = 16, normalized size = 0.8 \[ -\frac{1}{4} \csc ^4(x)+\csc ^2(x)+\log (\sin (x)) \]
Antiderivative was successfully verified.
[In] Integrate[Cot[x]^5,x]
[Out]
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Maple [A] time = 0.01, size = 26, normalized size = 1.3 \[ -{\frac{\ln \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{2}}-{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}+\ln \left ( \tan \left ( x \right ) \right ) +{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/tan(x)^5,x)
[Out]
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Maxima [A] time = 1.44574, size = 30, normalized size = 1.5 \[ \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^(-5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214313, size = 54, normalized size = 2.7 \[ \frac{2 \, \log \left (\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \tan \left (x\right )^{4} + 3 \, \tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} - 1}{4 \, \tan \left (x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^(-5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.114924, size = 19, normalized size = 0.95 \[ \frac{4 \sin ^{2}{\left (x \right )} - 1}{4 \sin ^{4}{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/tan(x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.208124, size = 50, normalized size = 2.5 \[ -\frac{3 \, \tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1}{4 \, \tan \left (x\right )^{4}} - \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(tan(x)^(-5),x, algorithm="giac")
[Out]