Optimal. Leaf size=41 \[ \tan (x)+\frac {5 \tan ^3(x)}{3}+2 \tan ^5(x)+\frac {10 \tan ^7(x)}{7}+\frac {5 \tan ^9(x)}{9}+\frac {\tan ^{11}(x)}{11} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852}
\begin {gather*} \frac {\tan ^{11}(x)}{11}+\frac {5 \tan ^9(x)}{9}+\frac {10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac {5 \tan ^3(x)}{3}+\tan (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \sec ^{12}(x) \, dx &=-\text {Subst}\left (\int \left (1+5 x^2+10 x^4+10 x^6+5 x^8+x^{10}\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac {5 \tan ^3(x)}{3}+2 \tan ^5(x)+\frac {10 \tan ^7(x)}{7}+\frac {5 \tan ^9(x)}{9}+\frac {\tan ^{11}(x)}{11}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 57, normalized size = 1.39 \begin {gather*} \frac {256 \tan (x)}{693}+\frac {128}{693} \sec ^2(x) \tan (x)+\frac {32}{231} \sec ^4(x) \tan (x)+\frac {80}{693} \sec ^6(x) \tan (x)+\frac {10}{99} \sec ^8(x) \tan (x)+\frac {1}{11} \sec ^{10}(x) \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 37, normalized size = 0.90
method | result | size |
default | \(-\left (-\frac {256}{693}-\frac {\left (\sec ^{10}\left (x \right )\right )}{11}-\frac {10 \left (\sec ^{8}\left (x \right )\right )}{99}-\frac {80 \left (\sec ^{6}\left (x \right )\right )}{693}-\frac {32 \left (\sec ^{4}\left (x \right )\right )}{231}-\frac {128 \left (\sec ^{2}\left (x \right )\right )}{693}\right ) \tan \left (x \right )\) | \(37\) |
risch | \(\frac {512 i \left (462 \,{\mathrm e}^{10 i x}+330 \,{\mathrm e}^{8 i x}+165 \,{\mathrm e}^{6 i x}+55 \,{\mathrm e}^{4 i x}+11 \,{\mathrm e}^{2 i x}+1\right )}{693 \left ({\mathrm e}^{2 i x}+1\right )^{11}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.94, size = 33, normalized size = 0.80 \begin {gather*} \frac {1}{11} \, \tan \left (x\right )^{11} + \frac {5}{9} \, \tan \left (x\right )^{9} + \frac {10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac {5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 40, normalized size = 0.98 \begin {gather*} \frac {{\left (256 \, \cos \left (x\right )^{10} + 128 \, \cos \left (x\right )^{8} + 96 \, \cos \left (x\right )^{6} + 80 \, \cos \left (x\right )^{4} + 70 \, \cos \left (x\right )^{2} + 63\right )} \sin \left (x\right )}{693 \, \cos \left (x\right )^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 66, normalized size = 1.61 \begin {gather*} \frac {256 \sin {\left (x \right )}}{693 \cos {\left (x \right )}} + \frac {128 \sin {\left (x \right )}}{693 \cos ^{3}{\left (x \right )}} + \frac {32 \sin {\left (x \right )}}{231 \cos ^{5}{\left (x \right )}} + \frac {80 \sin {\left (x \right )}}{693 \cos ^{7}{\left (x \right )}} + \frac {10 \sin {\left (x \right )}}{99 \cos ^{9}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{11 \cos ^{11}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 33, normalized size = 0.80 \begin {gather*} \frac {1}{11} \, \tan \left (x\right )^{11} + \frac {5}{9} \, \tan \left (x\right )^{9} + \frac {10}{7} \, \tan \left (x\right )^{7} + 2 \, \tan \left (x\right )^{5} + \frac {5}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 51, normalized size = 1.24 \begin {gather*} \frac {256\,\sin \left (x\right )\,{\cos \left (x\right )}^{10}+128\,\sin \left (x\right )\,{\cos \left (x\right )}^8+96\,\sin \left (x\right )\,{\cos \left (x\right )}^6+80\,\sin \left (x\right )\,{\cos \left (x\right )}^4+70\,\sin \left (x\right )\,{\cos \left (x\right )}^2+63\,\sin \left (x\right )}{693\,{\cos \left (x\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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