Optimal. Leaf size=20 \[ 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3396}
\begin {gather*} 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\cos ^{\frac {3}{2}}(x)}+x \sqrt {\cos (x)}\right ) \, dx &=\int \frac {x}{\cos ^{\frac {3}{2}}(x)} \, dx+\int x \sqrt {\cos (x)} \, dx\\ &=4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 17, normalized size = 0.85 \begin {gather*} \frac {2 (2 \cos (x)+x \sin (x))}{\sqrt {\cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (x \right )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (\cos ^{2}{\left (x \right )} + 1\right )}{\cos ^{\frac {3}{2}}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 15, normalized size = 0.75 \begin {gather*} \frac {4\,\cos \left (x\right )+2\,x\,\sin \left (x\right )}{\sqrt {\cos \left (x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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