Optimal. Leaf size=41 \[ \frac {\tan ^{11}(x)}{11}+\frac {5 \tan ^9(x)}{9}+\frac {10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac {5 \tan ^3(x)}{3}+\tan (x) \]
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Rubi [A] time = 0.02, 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 = {3767} \[ \frac {\tan ^{11}(x)}{11}+\frac {5 \tan ^9(x)}{9}+\frac {10 \tan ^7(x)}{7}+2 \tan ^5(x)+\frac {5 \tan ^3(x)}{3}+\tan (x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \sec ^{12}(x) \, dx &=-\operatorname {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 \[ \frac {256 \tan (x)}{693}+\frac {1}{11} \tan (x) \sec ^{10}(x)+\frac {10}{99} \tan (x) \sec ^8(x)+\frac {80}{693} \tan (x) \sec ^6(x)+\frac {32}{231} \tan (x) \sec ^4(x)+\frac {128}{693} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^{12}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 40, normalized size = 0.98 \[ \frac {{\left (256 \, \cos \relax (x)^{10} + 128 \, \cos \relax (x)^{8} + 96 \, \cos \relax (x)^{6} + 80 \, \cos \relax (x)^{4} + 70 \, \cos \relax (x)^{2} + 63\right )} \sin \relax (x)}{693 \, \cos \relax (x)^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 33, normalized size = 0.80 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {5}{9} \, \tan \relax (x)^{9} + \frac {10}{7} \, \tan \relax (x)^{7} + 2 \, \tan \relax (x)^{5} + \frac {5}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 37, normalized size = 0.90
method | result | size |
default | \(-\left (-\frac {256}{693}-\frac {\left (\sec ^{10}\relax (x )\right )}{11}-\frac {10 \left (\sec ^{8}\relax (x )\right )}{99}-\frac {80 \left (\sec ^{6}\relax (x )\right )}{693}-\frac {32 \left (\sec ^{4}\relax (x )\right )}{231}-\frac {128 \left (\sec ^{2}\relax (x )\right )}{693}\right ) \tan \relax (x )\) | \(37\) |
risch | \(\frac {512 i \left (462 \,{\mathrm e}^{10 i x}+330 \,{\mathrm e}^{8 i x}+165 \,{\mathrm e}^{6 i x}+55 \,{\mathrm e}^{4 i x}+11 \,{\mathrm e}^{2 i x}+1\right )}{693 \left (1+{\mathrm e}^{2 i x}\right )^{11}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 33, normalized size = 0.80 \[ \frac {1}{11} \, \tan \relax (x)^{11} + \frac {5}{9} \, \tan \relax (x)^{9} + \frac {10}{7} \, \tan \relax (x)^{7} + 2 \, \tan \relax (x)^{5} + \frac {5}{3} \, \tan \relax (x)^{3} + \tan \relax (x) \]
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 \[ \frac {256\,\sin \relax (x)\,{\cos \relax (x)}^{10}+128\,\sin \relax (x)\,{\cos \relax (x)}^8+96\,\sin \relax (x)\,{\cos \relax (x)}^6+80\,\sin \relax (x)\,{\cos \relax (x)}^4+70\,\sin \relax (x)\,{\cos \relax (x)}^2+63\,\sin \relax (x)}{693\,{\cos \relax (x)}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 66, normalized size = 1.61 \[ \frac {256 \sin {\relax (x )}}{693 \cos {\relax (x )}} + \frac {128 \sin {\relax (x )}}{693 \cos ^{3}{\relax (x )}} + \frac {32 \sin {\relax (x )}}{231 \cos ^{5}{\relax (x )}} + \frac {80 \sin {\relax (x )}}{693 \cos ^{7}{\relax (x )}} + \frac {10 \sin {\relax (x )}}{99 \cos ^{9}{\relax (x )}} + \frac {\sin {\relax (x )}}{11 \cos ^{11}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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