Optimal. Leaf size=20 \[ 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \]
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Rubi [A] time = 0.04, 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 = {3315} \[ 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
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 = 0.09, size = 17, normalized size = 0.85 \[ \frac {2 (x \sin (x)+2 \cos (x))}{\sqrt {\cos (x)}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \relax (x )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\relax (x )\right )\, dx \]
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 \[ \int x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ \frac {4\,\cos \relax (x)+2\,x\,\sin \relax (x)}{\sqrt {\cos \relax (x)}} \]
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 \[ \int \frac {x \left (\cos ^{2}{\relax (x )} + 1\right )}{\cos ^{\frac {3}{2}}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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