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 = 0.0280991, antiderivative size = 46, normalized size of antiderivative = 1., 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 2590
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0149302, size = 46, normalized size = 1. \[ \frac{\cos ^{10}(x)}{10}-\frac{5 \cos ^8(x)}{8}+\frac{5 \cos ^6(x)}{3}-\frac{5 \cos ^4(x)}{2}+\frac{5 \cos ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 37, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{10}}{10}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{8}}{8}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{6}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}}-\ln \left ( \cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935469, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{10} \, \sin \left (x\right )^{10} - \frac{1}{8} \, \sin \left (x\right )^{8} - \frac{1}{6} \, \sin \left (x\right )^{6} - \frac{1}{4} \, \sin \left (x\right )^{4} - \frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12827, size = 123, normalized size = 2.67 \begin{align*} \frac{1}{10} \, \cos \left (x\right )^{10} - \frac{5}{8} \, \cos \left (x\right )^{8} + \frac{5}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} - \log \left (-\cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090494, size = 44, normalized size = 0.96 \begin{align*} - \log{\left (\cos{\left (x \right )} \right )} + \frac{\cos ^{10}{\left (x \right )}}{10} - \frac{5 \cos ^{8}{\left (x \right )}}{8} + \frac{5 \cos ^{6}{\left (x \right )}}{3} - \frac{5 \cos ^{4}{\left (x \right )}}{2} + \frac{5 \cos ^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07015, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{10} \, \cos \left (x\right )^{10} - \frac{5}{8} \, \cos \left (x\right )^{8} + \frac{5}{3} \, \cos \left (x\right )^{6} - \frac{5}{2} \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} - \frac{1}{2} \, \log \left (\cos \left (x\right )^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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