Optimal. Leaf size=16 \[ -\frac {2}{\sin (x)+1}-\log (\sin (x)+1) \]
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Rubi [A] time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4391, 2667, 43} \[ -\frac {2}{\sin (x)+1}-\log (\sin (x)+1) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4391
Rubi steps
\begin {align*} \int \frac {1}{(\sec (x)+\tan (x))^3} \, dx &=\int \frac {\cos ^3(x)}{(1+\sin (x))^3} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,\sin (x)\right )\\ &=-\log (1+\sin (x))-\frac {2}{1+\sin (x)}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 34, normalized size = 2.12 \[ -\frac {2}{\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2}-2 \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 20, normalized size = 1.25 \[ -\frac {{\left (\sin \relax (x) + 1\right )} \log \left (\sin \relax (x) + 1\right ) + 2}{\sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 45, normalized size = 2.81 \[ \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, x\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{2}} + \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 17, normalized size = 1.06 \[ -\ln \left (1+\sin \relax (x )\right )-\frac {2}{1+\sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 64, normalized size = 4.00 \[ \frac {4 \, \sin \relax (x)}{{\left (\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )} {\left (\cos \relax (x) + 1\right )}} - 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 41, normalized size = 2.56 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )+\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.83, size = 301, normalized size = 18.81 \[ - \frac {2 \log {\left (\tan {\relax (x )} + \sec {\relax (x )} \right )} \tan ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} - \frac {4 \log {\left (\tan {\relax (x )} + \sec {\relax (x )} \right )} \tan {\relax (x )} \sec {\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} - \frac {2 \log {\left (\tan {\relax (x )} + \sec {\relax (x )} \right )} \sec ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} + \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )} \tan ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} + \frac {2 \log {\left (\tan ^{2}{\relax (x )} + 1 \right )} \tan {\relax (x )} \sec {\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} + \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )} \sec ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} + \frac {\tan ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} - \frac {\sec ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} - \frac {1}{2 \tan ^{2}{\relax (x )} + 4 \tan {\relax (x )} \sec {\relax (x )} + 2 \sec ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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