Optimal. Leaf size=91 \[ -\frac{3 \sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}+\frac{3 \sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}} \]
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Rubi [A] time = 0.120849, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.909, Rules used = {4397, 4400, 2597, 2599, 2601, 2565, 329, 298, 203, 206} \[ -\frac{3 \sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}+\frac{3 \sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4400
Rule 2597
Rule 2599
Rule 2601
Rule 2565
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(-\cos (x)+\sec (x))^{5/2}} \, dx &=\int \frac{1}{(\sin (x) \tan (x))^{5/2}} \, dx\\ &=\frac{\left (\sqrt{\sin (x)} \sqrt{\tan (x)}\right ) \int \frac{1}{\sin ^{\frac{5}{2}}(x) \tan ^{\frac{5}{2}}(x)} \, dx}{\sqrt{\sin (x) \tan (x)}}\\ &=-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}-\frac{\left (3 \sqrt{\sin (x)} \sqrt{\tan (x)}\right ) \int \frac{1}{\sin ^{\frac{5}{2}}(x) \sqrt{\tan (x)}} \, dx}{8 \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}-\frac{\left (3 \sqrt{\sin (x)} \sqrt{\tan (x)}\right ) \int \frac{1}{\sqrt{\sin (x)} \sqrt{\tan (x)}} \, dx}{32 \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}-\frac{(3 \sin (x)) \int \sqrt{\cos (x)} \csc (x) \, dx}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}+\frac{(3 \sin (x)) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-x^2} \, dx,x,\cos (x)\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}+\frac{(3 \sin (x)) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{\cos (x)}\right )}{16 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}+\frac{(3 \sin (x)) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{(3 \sin (x)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\cos (x)}\right )}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}-\frac{3 \tan ^{-1}\left (\sqrt{\cos (x)}\right ) \sin (x)}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{3 \tanh ^{-1}\left (\sqrt{\cos (x)}\right ) \sin (x)}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ \end{align*}
Mathematica [A] time = 0.647308, size = 73, normalized size = 0.8 \[ -\frac{\cot (x) \sqrt{\sin (x) \tan (x)} \left (3 \cos (x) \tan ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )-3 \cos (x) \tanh ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )+\cos ^2(x)^{3/4} (3 \cos (2 x)+5) \cot (x) \csc ^3(x)\right )}{32 \cos ^2(x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.15, size = 454, normalized size = 5. \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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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Fricas [B] time = 2.47547, size = 452, normalized size = 4.97 \begin{align*} \frac{3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \arctan \left (\frac{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 4 \,{\left (3 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{64 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\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 \frac{1}{{\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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