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 = 0.06, 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 = {3315} \[ -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
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 = 0.14, size = 33, normalized size = 0.70 \[ \frac {21 x \sin (x)+9 x \sin (3 x)+46 \cos (x)+18 \cos (3 x)}{30 \cos ^{\frac {5}{2}}(x)} \]
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 \[ \int \frac {3}{5} \, x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \relax (x )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cos }\relax (x )\right )}{5}\, dx \]
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 \[ \int \frac {3}{5} \, x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {7}{2}}}\,{d x} \]
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 \[ \frac {36\,{\cos \relax (x)}^3+18\,x\,\sin \relax (x)\,{\cos \relax (x)}^2-4\,\cos \relax (x)+6\,x\,\sin \relax (x)}{15\,{\cos \relax (x)}^{5/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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