Optimal. Leaf size=73 \[ -\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{256}{35} \csc (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{35} \tan (x) \sec (x) \sqrt{\sin (x) \tan (x)} \]
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Rubi [A] time = 0.112532, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{256}{35} \csc (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{35} \tan (x) \sec (x) \sqrt{\sin (x) \tan (x)} \]
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))^{7/2} \, dx &=\int (\sin (x) \tan (x))^{7/2} \, dx\\ &=\frac{\sqrt{\sin (x) \tan (x)} \int \sin ^{\frac{7}{2}}(x) \tan ^{\frac{7}{2}}(x) \, dx}{\sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}+\frac{\left (12 \sqrt{\sin (x) \tan (x)}\right ) \int \sin ^{\frac{3}{2}}(x) \tan ^{\frac{7}{2}}(x) \, dx}{7 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}+\frac{\left (32 \sqrt{\sin (x) \tan (x)}\right ) \int \frac{\tan ^{\frac{7}{2}}(x)}{\sqrt{\sin (x)}} \, dx}{7 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=\frac{64}{35} \sec (x) \tan (x) \sqrt{\sin (x) \tan (x)}-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{\left (128 \sqrt{\sin (x) \tan (x)}\right ) \int \frac{\tan ^{\frac{3}{2}}(x)}{\sqrt{\sin (x)}} \, dx}{35 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-\frac{256}{35} \csc (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{35} \sec (x) \tan (x) \sqrt{\sin (x) \tan (x)}-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}\\ \end{align*}
Mathematica [A] time = 0.204435, size = 37, normalized size = 0.51 \[ \frac{1}{70} \sec (x) \sqrt{\sin (x) \tan (x)} (28 \tan (x)-512 \cot (x)-5 (\sin (3 x)-23 \sin (x)) \cos (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.235, size = 603, normalized size = 8.3 \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 [A] time = 1.57384, size = 111, normalized size = 1.52 \begin{align*} \frac{128 \,{\left (\frac{7 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{7 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac{2 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - 2\right )}}{35 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1244, size = 134, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (5 \, \cos \left (x\right )^{6} - 35 \, \cos \left (x\right )^{4} - 105 \, \cos \left (x\right )^{2} + 7\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{35 \, \cos \left (x\right )^{2} \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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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