Integrand size = 11, antiderivative size = 55 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{8 \sqrt {2}}+\frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x) \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4442, 201, 223, 212} \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{8 \sqrt {2}}-\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)+\frac {3}{8} \cos (x) \sqrt {\cos (2 x)} \]
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Rule 201
Rule 212
Rule 223
Rule 4442
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (-1+2 x^2\right )^{3/2} \, dx,x,\cos (x)\right ) \\ & = -\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)+\frac {3}{4} \text {Subst}\left (\int \sqrt {-1+2 x^2} \, dx,x,\cos (x)\right ) \\ & = \frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)-\frac {3}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,\cos (x)\right ) \\ & = \frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)-\frac {3}{8} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right ) \\ & = -\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{8 \sqrt {2}}+\frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {1}{8} \sqrt {\cos (2 x)} (-2 \cos (x)+\cos (3 x))-\frac {3 \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{8 \sqrt {2}} \]
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Time = 0.40 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {3 \ln \left (\cos \left (x \right ) \sqrt {2}+\sqrt {2 \left (\cos ^{2}\left (x \right )\right )-1}\right ) \sqrt {2}}{16}-\frac {\left (\cos ^{3}\left (x \right )\right ) \sqrt {2 \left (\cos ^{2}\left (x \right )\right )-1}}{2}+\frac {5 \cos \left (x \right ) \sqrt {2 \left (\cos ^{2}\left (x \right )\right )-1}}{8}\) | \(55\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.87 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {1}{8} \, {\left (4 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1} + \frac {3}{128} \, \sqrt {2} \log \left (2048 \, \cos \left (x\right )^{8} - 2048 \, \cos \left (x\right )^{6} + 640 \, \cos \left (x\right )^{4} - 64 \, \cos \left (x\right )^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \left (x\right )^{7} - 96 \, \sqrt {2} \cos \left (x\right )^{5} + 20 \, \sqrt {2} \cos \left (x\right )^{3} - \sqrt {2} \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1} + 1\right ) \]
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\[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=\int \sin {\left (x \right )} \cos ^{\frac {3}{2}}{\left (2 x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (39) = 78\).
Time = 0.34 (sec) , antiderivative size = 790, normalized size of antiderivative = 14.36 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=\text {Too large to display} \]
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none
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {1}{8} \, {\left (4 \, \cos \left (x\right )^{2} - 5\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1} \cos \left (x\right ) + \frac {3}{16} \, \sqrt {2} \log \left ({\left | -\sqrt {2} \cos \left (x\right ) + \sqrt {2 \, \cos \left (x\right )^{2} - 1} \right |}\right ) \]
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Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx=-\frac {{\cos \left (2\,x\right )}^{3/2}\,\cos \left (x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ \cos \left (2\,x\right )+1\right )}{{\left (-\cos \left (2\,x\right )\right )}^{3/2}} \]
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