Optimal. Leaf size=26 \[ \frac{3}{8} \tanh ^{-1}(\sin (x))+\frac{1}{4} \tan (x) \sec ^3(x)+\frac{3}{8} \tan (x) \sec (x) \]
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Rubi [A] time = 0.0143809, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3768, 3770} \[ \frac{3}{8} \tanh ^{-1}(\sin (x))+\frac{1}{4} \tan (x) \sec ^3(x)+\frac{3}{8} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^5(x) \, dx &=\frac{1}{4} \sec ^3(x) \tan (x)+\frac{3}{4} \int \sec ^3(x) \, dx\\ &=\frac{3}{8} \sec (x) \tan (x)+\frac{1}{4} \sec ^3(x) \tan (x)+\frac{3}{8} \int \sec (x) \, dx\\ &=\frac{3}{8} \tanh ^{-1}(\sin (x))+\frac{3}{8} \sec (x) \tan (x)+\frac{1}{4} \sec ^3(x) \tan (x)\\ \end{align*}
Mathematica [B] time = 0.113231, size = 58, normalized size = 2.23 \[ \frac{1}{16} \left (\frac{1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+6 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 25, normalized size = 1. \begin{align*} - \left ( -{\frac{ \left ( \sec \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\sec \left ( x \right ) }{8}} \right ) \tan \left ( x \right ) +{\frac{3\,\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.946055, size = 57, normalized size = 2.19 \begin{align*} -\frac{3 \, \sin \left (x\right )^{3} - 5 \, \sin \left (x\right )}{8 \,{\left (\sin \left (x\right )^{4} - 2 \, \sin \left (x\right )^{2} + 1\right )}} + \frac{3}{16} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{3}{16} \, \log \left (\sin \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16144, size = 138, normalized size = 5.31 \begin{align*} \frac{3 \, \cos \left (x\right )^{4} \log \left (\sin \left (x\right ) + 1\right ) - 3 \, \cos \left (x\right )^{4} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \,{\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{16 \, \cos \left (x\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.135195, size = 46, normalized size = 1.77 \begin{align*} - \frac{3 \sin ^{3}{\left (x \right )} - 5 \sin{\left (x \right )}}{8 \sin ^{4}{\left (x \right )} - 16 \sin ^{2}{\left (x \right )} + 8} - \frac{3 \log{\left (\sin{\left (x \right )} - 1 \right )}}{16} + \frac{3 \log{\left (\sin{\left (x \right )} + 1 \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08199, size = 51, normalized size = 1.96 \begin{align*} -\frac{3 \, \sin \left (x\right )^{3} - 5 \, \sin \left (x\right )}{8 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac{3}{16} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{3}{16} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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