Optimal. Leaf size=19 \[ \frac{2}{3} \sin (x) \cos (x) \sqrt{\tan (x) \sec ^4(x)} \]
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Rubi [A] time = 0.123945, antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1999, 1954, 1250, 30} \[ \frac{2 \tan ^2(x) \sec ^2(x)}{3 \sqrt{\tan ^5(x)+2 \tan ^3(x)+\tan (x)}} \]
Antiderivative was successfully verified.
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Rule 1999
Rule 1954
Rule 1250
Rule 30
Rubi steps
\begin{align*} \int \sqrt{\sec ^4(x) \tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{\sqrt{x \left (1+x^2\right )^2}} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{\sqrt{x+2 x^3+x^5}} \, dx,x,\tan (x)\right )\\ &=\frac{\left (\sqrt{\tan (x)} \sqrt{1+2 \tan ^2(x)+\tan ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (1+x^2\right )}{\sqrt{1+2 x^2+x^4}} \, dx,x,\tan (x)\right )}{\sqrt{\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ &=\frac{\left (\sec ^2(x) \sqrt{\tan (x)}\right ) \operatorname{Subst}\left (\int \sqrt{x} \, dx,x,\tan (x)\right )}{\sqrt{\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ &=\frac{2 \sec ^2(x) \tan ^2(x)}{3 \sqrt{\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ \end{align*}
Mathematica [A] time = 0.0074824, size = 19, normalized size = 1. \[ \frac{2}{3} \sin (x) \cos (x) \sqrt{\tan (x) \sec ^4(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,\cos \left ( x \right ) \sin \left ( x \right ) }{3}\sqrt{{\frac{\sin \left ( x \right ) }{ \left ( \cos \left ( x \right ) \right ) ^{5}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49072, size = 8, normalized size = 0.42 \begin{align*} \frac{2}{3} \, \tan \left (x\right )^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12696, size = 55, normalized size = 2.89 \begin{align*} \frac{2}{3} \, \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right )^{5}}} \cos \left (x\right ) \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right )^{5}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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