Optimal. Leaf size=73 \[ \frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}-\frac{64}{35} \cot (x) \csc (x) \sqrt{\cos (x) \cot (x)}+\frac{256}{35} \sec (x) \sqrt{\cos (x) \cot (x)} \]
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Rubi [A] time = 0.148391, 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} \[ \frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}-\frac{64}{35} \cot (x) \csc (x) \sqrt{\cos (x) \cot (x)}+\frac{256}{35} \sec (x) \sqrt{\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4400
Rule 2598
Rule 2594
Rule 2589
Rubi steps
\begin{align*} \int (\csc (x)-\sin (x))^{7/2} \, dx &=\int (\cos (x) \cot (x))^{7/2} \, dx\\ &=\frac{\sqrt{\cos (x) \cot (x)} \int \cos ^{\frac{7}{2}}(x) \cot ^{\frac{7}{2}}(x) \, dx}{\sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{\left (12 \sqrt{\cos (x) \cot (x)}\right ) \int \cos ^{\frac{3}{2}}(x) \cot ^{\frac{7}{2}}(x) \, dx}{7 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{\left (32 \sqrt{\cos (x) \cot (x)}\right ) \int \frac{\cot ^{\frac{7}{2}}(x)}{\sqrt{\cos (x)}} \, dx}{7 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}-\frac{64}{35} \cot (x) \sqrt{\cos (x) \cot (x)} \csc (x)-\frac{\left (128 \sqrt{\cos (x) \cot (x)}\right ) \int \frac{\cot ^{\frac{3}{2}}(x)}{\sqrt{\cos (x)}} \, dx}{35 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}-\frac{64}{35} \cot (x) \sqrt{\cos (x) \cot (x)} \csc (x)+\frac{256}{35} \sqrt{\cos (x) \cot (x)} \sec (x)\\ \end{align*}
Mathematica [A] time = 0.0795734, size = 37, normalized size = 0.51 \[ -\frac{1}{70} \sec (x) \sqrt{\cos (x) \cot (x)} \left (115 \cos ^2(x)+5 \cos (3 x) \cos (x)+28 \cot ^2(x)-512\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.157, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( 10\, \left ( \cos \left ( x \right ) \right ) ^{6}+40\, \left ( \cos \left ( x \right ) \right ) ^{4}-320\, \left ( \cos \left ( x \right ) \right ) ^{2}+256 \right ) \sin \left ( x \right ) }{35\, \left ( \cos \left ( x \right ) \right ) ^{7}} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.97216, size = 780, normalized size = 10.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33403, size = 131, normalized size = 1.79 \begin{align*} -\frac{2 \,{\left (5 \, \cos \left (x\right )^{6} + 20 \, \cos \left (x\right )^{4} - 160 \, \cos \left (x\right )^{2} + 128\right )} \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{35 \,{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\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 (\csc \left (x\right ) - \sin \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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