Optimal. Leaf size=32 \[ -x \tan ^2(x) \sqrt{a \cot ^4(x)}-\tan (x) \sqrt{a \cot ^4(x)} \]
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Rubi [A] time = 0.0154425, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 8} \[ -x \tan ^2(x) \sqrt{a \cot ^4(x)}-\tan (x) \sqrt{a \cot ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \cot ^4(x)} \, dx &=\left (\sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^2(x) \, dx\\ &=-\sqrt{a \cot ^4(x)} \tan (x)-\left (\sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int 1 \, dx\\ &=-\sqrt{a \cot ^4(x)} \tan (x)-x \sqrt{a \cot ^4(x)} \tan ^2(x)\\ \end{align*}
Mathematica [A] time = 0.0153746, size = 20, normalized size = 0.62 \[ \tan ^2(x) (x+\cot (x)) \left (-\sqrt{a \cot ^4(x)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 27, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( \cot \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{4}} \left ( -\cot \left ( x \right ) +{\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( x \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64365, size = 22, normalized size = 0.69 \begin{align*} -\sqrt{a} x - \frac{\sqrt{a}}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15884, size = 154, normalized size = 4.81 \begin{align*} \frac{{\left (x \cos \left (2 \, x\right ) - x - \sin \left (2 \, x\right )\right )} \sqrt{\frac{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{\cos \left (2 \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cot ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29934, size = 28, normalized size = 0.88 \begin{align*} -\frac{1}{2} \, \sqrt{a}{\left (2 \, x + \frac{1}{\tan \left (\frac{1}{2} \, x\right )} - \tan \left (\frac{1}{2} \, x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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