Optimal. Leaf size=47 \[ -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396}
\begin {gather*} -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \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 {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\cos (x)} \, dx+\int \frac {x}{\cos ^{\frac {7}{2}}(x)} \, dx\\ &=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {3}{5} \int \frac {x}{\cos ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\cos (x)} \, dx\\ &=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 33, normalized size = 0.70 \begin {gather*} \frac {46 \cos (x)+18 \cos (3 x)+21 x \sin (x)+9 x \sin (3 x)}{30 \cos ^{\frac {5}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cos }\left (x \right )\right )}{5}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 0.53, size = 31, normalized size = 0.66 \begin {gather*} \frac {36\,{\cos \left (x\right )}^3+18\,x\,\sin \left (x\right )\,{\cos \left (x\right )}^2-4\,\cos \left (x\right )+6\,x\,\sin \left (x\right )}{15\,{\cos \left (x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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