Optimal. Leaf size=50 \[ -\frac {\cos (x)-\sin (x)}{15 (\sin (x)+\cos (x))^3}-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}+\frac {2 \sin (x)}{15 (\sin (x)+\cos (x))} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3076, 3075} \[ -\frac {\cos (x)-\sin (x)}{15 (\sin (x)+\cos (x))^3}-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}+\frac {2 \sin (x)}{15 (\sin (x)+\cos (x))} \]
Antiderivative was successfully verified.
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Rule 3075
Rule 3076
Rubi steps
\begin {align*} \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx &=-\frac {\cos (x)-\sin (x)}{10 (\cos (x)+\sin (x))^5}+\frac {2}{5} \int \frac {1}{(\cos (x)+\sin (x))^4} \, dx\\ &=-\frac {\cos (x)-\sin (x)}{10 (\cos (x)+\sin (x))^5}-\frac {\cos (x)-\sin (x)}{15 (\cos (x)+\sin (x))^3}+\frac {2}{15} \int \frac {1}{(\cos (x)+\sin (x))^2} \, dx\\ &=-\frac {\cos (x)-\sin (x)}{10 (\cos (x)+\sin (x))^5}-\frac {\cos (x)-\sin (x)}{15 (\cos (x)+\sin (x))^3}+\frac {2 \sin (x)}{15 (\cos (x)+\sin (x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.52 \[ -\frac {-10 \sin (x)+\sin (5 x)+5 \cos (3 x)}{30 (\sin (x)+\cos (x))^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 67, normalized size = 1.34 \[ -\frac {8 \, \cos \relax (x)^{5} - 20 \, \cos \relax (x)^{3} - {\left (8 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{2} - 7\right )} \sin \relax (x) + 5 \, \cos \relax (x)}{30 \, {\left (4 \, \cos \relax (x)^{5} + {\left (4 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x) - 5 \, \cos \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 0.64 \[ -\frac {15 \, \tan \relax (x)^{4} + 30 \, \tan \relax (x)^{3} + 40 \, \tan \relax (x)^{2} + 20 \, \tan \relax (x) + 7}{15 \, {\left (\tan \relax (x) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 42, normalized size = 0.84 \[ \frac {2}{\left (\tan \relax (x )+1\right )^{4}}-\frac {4}{5 \left (\tan \relax (x )+1\right )^{5}}-\frac {8}{3 \left (\tan \relax (x )+1\right )^{3}}+\frac {2}{\left (\tan \relax (x )+1\right )^{2}}-\frac {1}{\tan \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 56, normalized size = 1.12 \[ -\frac {15 \, \tan \relax (x)^{4} + 30 \, \tan \relax (x)^{3} + 40 \, \tan \relax (x)^{2} + 20 \, \tan \relax (x) + 7}{15 \, {\left (\tan \relax (x)^{5} + 5 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{3} + 10 \, \tan \relax (x)^{2} + 5 \, \tan \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 88, normalized size = 1.76 \[ \frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-60\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+100\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5-118\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+100\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+60\,\mathrm {tan}\left (\frac {x}{2}\right )+15\right )}{15\,{\left (-{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.67, size = 838, normalized size = 16.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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