Optimal. Leaf size=4 \[ \cos (\cos (\sin (x))) \]
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Rubi [A]
time = 0.01, antiderivative size = 4, 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 = {4419, 4420,
2718} \begin {gather*} \cos (\cos (\sin (x))) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 4419
Rule 4420
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}(\int \sin (x) \sin (\cos (x)) \, dx,x,\sin (x))\\ &=-\text {Subst}(\int \sin (x) \, dx,x,\cos (\sin (x)))\\ &=\cos (\cos (\sin (x)))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 6.25, size = 4, normalized size = 1.00 \begin {gather*} \cos (\cos (\sin (x))) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.48, size = 5, normalized size = 1.25
method | result | size |
derivativedivides | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
default | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
risch | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
parallelrisch | \(-1+\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(7\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\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. 44 vs.
\(2 (4) = 8\).
time = 0.62, size = 44, normalized size = 11.00 \begin {gather*} \cos \left (\frac {\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} - 1}{\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.11, size = 5, normalized size = 1.25 \begin {gather*} \cos {\left (\cos {\left (\sin {\left (x \right )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}
Warning: Unable to verify antiderivative.
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