Optimal. Leaf size=22 \[ \frac {4}{\sin (x)+1}-\frac {2}{(\sin (x)+1)^2}+\log (\sin (x)+1) \]
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Rubi [A] time = 0.05, antiderivative size = 22, 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} \[ \frac {4}{\sin (x)+1}-\frac {2}{(\sin (x)+1)^2}+\log (\sin (x)+1) \]
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))^5} \, dx &=\int \frac {\cos ^5(x)}{(1+\sin (x))^5} \, dx\\ &=\operatorname {Subst}\left (\int \frac {(1-x)^2}{(1+x)^3} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {4}{(1+x)^3}-\frac {4}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx,x,\sin (x)\right )\\ &=\log (1+\sin (x))-\frac {2}{(1+\sin (x))^2}+\frac {4}{1+\sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 1.77 \[ \frac {4 \sin (x)+2}{\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4}+2 \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 35, normalized size = 1.59 \[ \frac {{\left (\cos \relax (x)^{2} - 2 \, \sin \relax (x) - 2\right )} \log \left (\sin \relax (x) + 1\right ) - 4 \, \sin \relax (x) - 2}{\cos \relax (x)^{2} - 2 \, \sin \relax (x) - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 64, normalized size = 2.91 \[ -\frac {25 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 100 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 198 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 100 \, \tan \left (\frac {1}{2} \, x\right ) + 25}{6 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{4}} - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 23, normalized size = 1.05 \[ \ln \left (1+\sin \relax (x )\right )-\frac {2}{\left (1+\sin \relax (x )\right )^{2}}+\frac {4}{1+\sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 92, normalized size = 4.18 \[ -\frac {8 \, \sin \relax (x)^{2}}{{\left (\frac {4 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {6 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {4 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {\sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 1\right )} {\left (\cos \relax (x) + 1\right )}^{2}} + 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) - \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 61, normalized size = 2.77 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-\frac {8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.76, size = 1059, normalized size = 48.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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