Optimal. Leaf size=46 \[ \frac {\cos ^{10}(x)}{10}-\frac {5 \cos ^8(x)}{8}+\frac {5 \cos ^6(x)}{3}-\frac {5 \cos ^4(x)}{2}+\frac {5 \cos ^2(x)}{2}-\log (\cos (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2590, 266, 43} \[ \frac {\cos ^{10}(x)}{10}-\frac {5 \cos ^8(x)}{8}+\frac {5 \cos ^6(x)}{3}-\frac {5 \cos ^4(x)}{2}+\frac {5 \cos ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2590
Rubi steps
\begin {align*} \int \sin ^{10}(x) \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^5}{x} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1-x)^5}{x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-5+\frac {1}{x}+10 x-10 x^2+5 x^3-x^4\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac {5 \cos ^2(x)}{2}-\frac {5 \cos ^4(x)}{2}+\frac {5 \cos ^6(x)}{3}-\frac {5 \cos ^8(x)}{8}+\frac {\cos ^{10}(x)}{10}-\log (\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ \frac {\cos ^{10}(x)}{10}-\frac {5 \cos ^8(x)}{8}+\frac {5 \cos ^6(x)}{3}-\frac {5 \cos ^4(x)}{2}+\frac {5 \cos ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{10}(x) \tan (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.95, size = 38, normalized size = 0.83 \[ \frac {1}{10} \, \cos \relax (x)^{10} - \frac {5}{8} \, \cos \relax (x)^{8} + \frac {5}{3} \, \cos \relax (x)^{6} - \frac {5}{2} \, \cos \relax (x)^{4} + \frac {5}{2} \, \cos \relax (x)^{2} - \log \left (-\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 42, normalized size = 0.91 \[ -\frac {1}{10} \, \sin \relax (x)^{10} - \frac {1}{8} \, \sin \relax (x)^{8} - \frac {1}{6} \, \sin \relax (x)^{6} - \frac {1}{4} \, \sin \relax (x)^{4} - \frac {1}{2} \, \sin \relax (x)^{2} - \frac {1}{2} \, \log \left (-\sin \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 37, normalized size = 0.80
method | result | size |
default | \(-\frac {\left (\sin ^{10}\relax (x )\right )}{10}-\frac {\left (\sin ^{8}\relax (x )\right )}{8}-\frac {\left (\sin ^{6}\relax (x )\right )}{6}-\frac {\left (\sin ^{4}\relax (x )\right )}{4}-\frac {\left (\sin ^{2}\relax (x )\right )}{2}-\ln \left (\cos \relax (x )\right )\) | \(37\) |
risch | \(i x +\frac {281 \,{\mathrm e}^{2 i x}}{1024}+\frac {281 \,{\mathrm e}^{-2 i x}}{1024}-\ln \left (1+{\mathrm e}^{2 i x}\right )+\frac {\cos \left (10 x \right )}{5120}-\frac {3 \cos \left (8 x \right )}{1024}+\frac {67 \cos \left (6 x \right )}{3072}-\frac {29 \cos \left (4 x \right )}{256}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 40, normalized size = 0.87 \[ -\frac {1}{10} \, \sin \relax (x)^{10} - \frac {1}{8} \, \sin \relax (x)^{8} - \frac {1}{6} \, \sin \relax (x)^{6} - \frac {1}{4} \, \sin \relax (x)^{4} - \frac {1}{2} \, \sin \relax (x)^{2} - \frac {1}{2} \, \log \left (\sin \relax (x)^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.78 \[ -\frac {{\sin \relax (x)}^{10}}{10}-\frac {{\sin \relax (x)}^8}{8}-\frac {{\sin \relax (x)}^6}{6}-\frac {{\sin \relax (x)}^4}{4}-\frac {{\sin \relax (x)}^2}{2}-\ln \left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 44, normalized size = 0.96 \[ - \log {\left (\cos {\relax (x )} \right )} + \frac {\cos ^{10}{\relax (x )}}{10} - \frac {5 \cos ^{8}{\relax (x )}}{8} + \frac {5 \cos ^{6}{\relax (x )}}{3} - \frac {5 \cos ^{4}{\relax (x )}}{2} + \frac {5 \cos ^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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