Optimal. Leaf size=21 \[ -\frac {2}{3} \cos ^{\frac {3}{2}}(x)+\frac {2}{7} \cos ^{\frac {7}{2}}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2645, 14}
\begin {gather*} \frac {2}{7} \cos ^{\frac {7}{2}}(x)-\frac {2}{3} \cos ^{\frac {3}{2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2645
Rubi steps
\begin {align*} \int \sqrt {\cos (x)} \sin ^3(x) \, dx &=-\text {Subst}\left (\int \sqrt {x} \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (\sqrt {x}-x^{5/2}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac {2}{3} \cos ^{\frac {3}{2}}(x)+\frac {2}{7} \cos ^{\frac {7}{2}}(x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 1.62 \begin {gather*} \frac {8 \sqrt [4]{\cos ^2(x)}+\cos ^2(x) (-11+3 \cos (2 x))}{21 \sqrt {\cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(13)=26\).
time = 0.06, size = 39, normalized size = 1.86
method | result | size |
default | \(-\frac {8 \sqrt {-2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+1}\, \left (6 \left (\sin ^{6}\left (\frac {x}{2}\right )\right )-9 \left (\sin ^{4}\left (\frac {x}{2}\right )\right )+\sin ^{2}\left (\frac {x}{2}\right )+1\right )}{21}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{7} \, \cos \left (x\right )^{\frac {7}{2}} - \frac {2}{3} \, \cos \left (x\right )^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{21} \, {\left (3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right )} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 29, normalized size = 1.38 \begin {gather*} -2 \left (-\frac {1}{7} \sqrt {\cos x} \cos ^{3}x+\frac {1}{3} \sqrt {\cos x} \cos x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 13, normalized size = 0.62 \begin {gather*} {\cos \left (x\right )}^{3/2}\,\left (\frac {2\,{\cos \left (x\right )}^2}{7}-\frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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