Optimal. Leaf size=32 \[ -\sqrt {a \cot ^4(x)} \tan (x)-x \sqrt {a \cot ^4(x)} \tan ^2(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8}
\begin {gather*} -x \tan ^2(x) \sqrt {a \cot ^4(x)}-\tan (x) \sqrt {a \cot ^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
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*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.62 \begin {gather*} -\sqrt {a \cot ^4(x)} (x+\cot (x)) \tan ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 27, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (\cot ^{4}\left (x \right )\right )}\, \left (-\cot \left (x \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\right )}{\cot \left (x \right )^{2}}\) | \(27\) |
default | \(\frac {\sqrt {a \left (\cot ^{4}\left (x \right )\right )}\, \left (-\cot \left (x \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\right )}{\cot \left (x \right )^{2}}\) | \(27\) |
risch | \(\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2} x}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {2 i \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 16, normalized size = 0.50 \begin {gather*} -\sqrt {a} x - \frac {\sqrt {a}}{\tan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (28) = 56\).
time = 3.31, size = 61, normalized size = 1.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \cot ^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 21, normalized size = 0.66 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {a\,{\mathrm {cot}\left (x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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