Optimal. Leaf size=50 \[ -\frac {\cos (x)-\sin (x)}{10 (\cos (x)+\sin (x))^5}-\frac {\cos (x)-\sin (x)}{15 (\cos (x)+\sin (x))^3}+\frac {2 \sin (x)}{15 (\cos (x)+\sin (x))} \]
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Rubi [A]
time = 0.02, 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 = {3155, 3154}
\begin {gather*} -\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))} \end {gather*}
Antiderivative was successfully verified.
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Rule 3154
Rule 3155
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 = 0.03, size = 26, normalized size = 0.52 \begin {gather*} -\frac {5 \cos (3 x)-10 \sin (x)+\sin (5 x)}{30 (\cos (x)+\sin (x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 42, normalized size = 0.84
method | result | size |
risch | \(\frac {-\frac {2}{15}+\frac {4 \,{\mathrm e}^{4 i x}}{3}+\frac {2 i {\mathrm e}^{2 i x}}{3}}{\left ({\mathrm e}^{2 i x}+i\right )^{5}}\) | \(30\) |
default | \(\frac {2}{\left (\tan \left (x \right )+1\right )^{2}}-\frac {8}{3 \left (\tan \left (x \right )+1\right )^{3}}-\frac {1}{\tan \left (x \right )+1}+\frac {2}{\left (\tan \left (x \right )+1\right )^{4}}-\frac {4}{5 \left (\tan \left (x \right )+1\right )^{5}}\) | \(42\) |
norman | \(\frac {-8 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right )-2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+8 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )-\frac {40 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {40 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3}-\frac {8 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}+\frac {8 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3}+\frac {236 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{15}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )-1\right )^{5}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.74, size = 56, normalized size = 1.12 \begin {gather*} -\frac {15 \, \tan \left (x\right )^{4} + 30 \, \tan \left (x\right )^{3} + 40 \, \tan \left (x\right )^{2} + 20 \, \tan \left (x\right ) + 7}{15 \, {\left (\tan \left (x\right )^{5} + 5 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{3} + 10 \, \tan \left (x\right )^{2} + 5 \, \tan \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 67, normalized size = 1.34 \begin {gather*} -\frac {8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} - {\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} - 7\right )} \sin \left (x\right ) + 5 \, \cos \left (x\right )}{30 \, {\left (4 \, \cos \left (x\right )^{5} + {\left (4 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 838 vs.
\(2 (51) = 102\).
time = 3.46, size = 838, normalized size = 16.76 \begin {gather*} - \frac {30 \tan ^{9}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} + \frac {120 \tan ^{8}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} - \frac {200 \tan ^{7}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} - \frac {40 \tan ^{6}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} + \frac {236 \tan ^{5}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} + \frac {40 \tan ^{4}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} - \frac {200 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} - \frac {120 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} - \frac {30 \tan {\left (\frac {x}{2} \right )}}{15 \tan ^{10}{\left (\frac {x}{2} \right )} - 150 \tan ^{9}{\left (\frac {x}{2} \right )} + 525 \tan ^{8}{\left (\frac {x}{2} \right )} - 600 \tan ^{7}{\left (\frac {x}{2} \right )} - 450 \tan ^{6}{\left (\frac {x}{2} \right )} + 1020 \tan ^{5}{\left (\frac {x}{2} \right )} + 450 \tan ^{4}{\left (\frac {x}{2} \right )} - 600 \tan ^{3}{\left (\frac {x}{2} \right )} - 525 \tan ^{2}{\left (\frac {x}{2} \right )} - 150 \tan {\left (\frac {x}{2} \right )} - 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 32, normalized size = 0.64 \begin {gather*} -\frac {15 \, \tan \left (x\right )^{4} + 30 \, \tan \left (x\right )^{3} + 40 \, \tan \left (x\right )^{2} + 20 \, \tan \left (x\right ) + 7}{15 \, {\left (\tan \left (x\right ) + 1\right )}^{5}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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