Optimal. Leaf size=11 \[ 6 \log (\cos (x))+2 \log (\sin (x)) \]
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Rubi [A]
time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4450, 457, 78}
\begin {gather*} 2 \log (\sin (x))+6 \log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 4450
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {2-8 x^2}{x \left (1-x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {2-8 x}{(1-x) x} \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {6}{-1+x}+\frac {2}{x}\right ) \, dx,x,\sin ^2(x)\right )\\ &=6 \log (\cos (x))+2 \log (\sin (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} 6 \log (\cos (x))+2 \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 12, normalized size = 1.09 \[-6 \ln \left (\tan \left (x \right )\right )+8 \ln \left (\sin \left (x \right )\right )\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (11) = 22\).
time = 0.43, size = 54, normalized size = 4.91 \begin {gather*} 3 \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 17, normalized size = 1.55 \begin {gather*} 3 \, \log \left (\cos \left (x\right )^{2}\right ) + \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 112.52, size = 15, normalized size = 1.36 \begin {gather*} 3 \log {\left (\sin ^{2}{\left (x \right )} - 1 \right )} + 2 \log {\left (\sin {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 16, normalized size = 1.45 \begin {gather*} \log \left (-\cos \left (x\right )^{2} + 1\right ) + 6 \, \log \left ({\left | \cos \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 11, normalized size = 1.00 \begin {gather*} 6\,\ln \left (\cos \left (x\right )\right )+\ln \left ({\sin \left (x\right )}^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 4, normalized size = 0.36 \begin {gather*} \frac {1}{\cos \left (x \right )^{2}} \end {gather*}
Warning: Unable to verify antiderivative.
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