Optimal. Leaf size=22 \[ -x+\frac {\tan ^5(x)}{5}-\frac {\tan ^3(x)}{3}+\tan (x) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ -x+\frac {\tan ^5(x)}{5}-\frac {\tan ^3(x)}{3}+\tan (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tan ^6(x) \, dx &=\frac {\tan ^5(x)}{5}-\int \tan ^4(x) \, dx\\ &=-\frac {1}{3} \tan ^3(x)+\frac {\tan ^5(x)}{5}+\int \tan ^2(x) \, dx\\ &=\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}-\int 1 \, dx\\ &=-x+\tan (x)-\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.36 \[ -x+\frac {23 \tan (x)}{15}+\frac {1}{5} \tan (x) \sec ^4(x)-\frac {11}{15} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \tan ^6(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.79, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.86
method | result | size |
norman | \(-x +\tan \relax (x )-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\frac {\left (\tan ^{5}\relax (x )\right )}{5}\) | \(19\) |
derivativedivides | \(\frac {\left (\tan ^{5}\relax (x )\right )}{5}-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )-\arctan \left (\tan \relax (x )\right )\) | \(21\) |
default | \(\frac {\left (\tan ^{5}\relax (x )\right )}{5}-\frac {\left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )-\arctan \left (\tan \relax (x )\right )\) | \(21\) |
risch | \(-x +\frac {2 i \left (45 \,{\mathrm e}^{8 i x}+90 \,{\mathrm e}^{6 i x}+140 \,{\mathrm e}^{4 i x}+70 \,{\mathrm e}^{2 i x}+23\right )}{15 \left (1+{\mathrm e}^{2 i x}\right )^{5}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 18, normalized size = 0.82 \[ \frac {1}{5} \, \tan \relax (x)^{5} - \frac {1}{3} \, \tan \relax (x)^{3} - x + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 18, normalized size = 0.82 \[ \frac {{\mathrm {tan}\relax (x)}^5}{5}-\frac {{\mathrm {tan}\relax (x)}^3}{3}+\mathrm {tan}\relax (x)-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 31, normalized size = 1.41 \[ - x + \frac {\sin ^{5}{\relax (x )}}{5 \cos ^{5}{\relax (x )}} - \frac {\sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} + \frac {\sin {\relax (x )}}{\cos {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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