Optimal. Leaf size=50 \[ -\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{15} \cot (x) \sqrt{\sin (x) \tan (x)} \]
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Rubi [A] time = 0.085911, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4397, 4400, 2598, 2594, 2589} \[ -\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{15} \cot (x) \sqrt{\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4400
Rule 2598
Rule 2594
Rule 2589
Rubi steps
\begin{align*} \int (-\cos (x)+\sec (x))^{5/2} \, dx &=\int (\sin (x) \tan (x))^{5/2} \, dx\\ &=\frac{\sqrt{\sin (x) \tan (x)} \int \sin ^{\frac{5}{2}}(x) \tan ^{\frac{5}{2}}(x) \, dx}{\sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{\left (8 \sqrt{\sin (x) \tan (x)}\right ) \int \sqrt{\sin (x)} \tan ^{\frac{5}{2}}(x) \, dx}{5 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}-\frac{\left (32 \sqrt{\sin (x) \tan (x)}\right ) \int \sqrt{\sin (x)} \sqrt{\tan (x)} \, dx}{15 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=\frac{64}{15} \cot (x) \sqrt{\sin (x) \tan (x)}+\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}\\ \end{align*}
Mathematica [A] time = 0.0763065, size = 29, normalized size = 0.58 \[ \frac{2}{15} \tan (x) \sqrt{\sin (x) \tan (x)} \left (3 \cos ^2(x)+32 \cot ^2(x)+5\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.146, size = 321, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58506, size = 111, normalized size = 2.22 \begin{align*} -\frac{32 \,{\left (\frac{5 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{5 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{2 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - 2\right )}}{15 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{5}{2}}{\left (-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{5}{2}}{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24485, size = 112, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (3 \, \cos \left (x\right )^{4} - 30 \, \cos \left (x\right )^{2} - 5\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{15 \, \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{\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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