Optimal. Leaf size=24 \[ -\frac {4}{3 \sqrt {\cos (x)}}+\frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396}
\begin {gather*} \frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}-\frac {4}{3 \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 {5}{2}}(x)}-\frac {x}{3 \sqrt {\cos (x)}}\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cos (x)}} \, dx\right )+\int \frac {x}{\cos ^{\frac {5}{2}}(x)} \, dx\\ &=-\frac {4}{3 \sqrt {\cos (x)}}+\frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 17, normalized size = 0.71 \begin {gather*} -\frac {8-4 x \tan (x)}{6 \sqrt {\cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (x \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cos \left (x \right )}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (x \sin \left (x\right ) - 2 \, \cos \left (x\right )\right )}}{3 \, \cos \left (x\right )^{\frac {3}{2}}} \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*} - \frac {\int \left (- \frac {3 x}{\cos ^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\cos {\left (x \right )}}}\, dx}{3} \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 = 0.14, size = 16, normalized size = 0.67 \begin {gather*} -\frac {4\,\cos \left (x\right )-2\,x\,\sin \left (x\right )}{3\,{\cos \left (x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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