Optimal. Leaf size=55 \[ -\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)+\frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4357, 195, 217, 206} \[ -\frac {1}{4} \cos (x) \cos ^{\frac {3}{2}}(2 x)+\frac {3}{8} \cos (x) \sqrt {\cos (2 x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4357
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx &=-\operatorname {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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )\\ &=-\frac {3 \tanh ^{-1}\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*}
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Mathematica [A] time = 0.09, size = 49, normalized size = 0.89 \[ -\frac {1}{8} \sqrt {\cos (2 x)} (\cos (3 x)-2 \cos (x))-\frac {3 \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{\frac {3}{2}}(2 x) \sin (x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.50, size = 103, normalized size = 1.87 \[ -\frac {1}{8} \, {\left (4 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1} + \frac {3}{128} \, \sqrt {2} \log \left (2048 \, \cos \relax (x)^{8} - 2048 \, \cos \relax (x)^{6} + 640 \, \cos \relax (x)^{4} - 64 \, \cos \relax (x)^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \relax (x)^{7} - 96 \, \sqrt {2} \cos \relax (x)^{5} + 20 \, \sqrt {2} \cos \relax (x)^{3} - \sqrt {2} \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 48, normalized size = 0.87 \[ -\frac {1}{8} \, {\left (4 \, \cos \relax (x)^{2} - 5\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1} \cos \relax (x) + \frac {3}{16} \, \sqrt {2} \log \left ({\left | -\sqrt {2} \cos \relax (x) + \sqrt {2 \, \cos \relax (x)^{2} - 1} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 55, normalized size = 1.00
method | result | size |
default | \(-\frac {\left (\cos ^{3}\relax (x )\right ) \sqrt {2 \left (\cos ^{2}\relax (x )\right )-1}}{2}+\frac {5 \cos \relax (x ) \sqrt {2 \left (\cos ^{2}\relax (x )\right )-1}}{8}-\frac {3 \ln \left (\cos \relax (x ) \sqrt {2}+\sqrt {2 \left (\cos ^{2}\relax (x )\right )-1}\right ) \sqrt {2}}{16}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.20, size = 790, normalized size = 14.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 29, normalized size = 0.53 \[ -\frac {{\cos \left (2\,x\right )}^{3/2}\,\cos \relax (x)\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ \cos \left (2\,x\right )+1\right )}{{\left (-\cos \left (2\,x\right )\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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