Optimal. Leaf size=42 \[ \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3391}
\begin {gather*} \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3391
Rubi steps
\begin {align*} \int \left (-\frac {x}{3 \sqrt {\cos (a+b x)}}+x \cos ^{\frac {3}{2}}(a+b x)\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cos (a+b x)}} \, dx\right )+\int x \cos ^{\frac {3}{2}}(a+b x) \, dx\\ &=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 40, normalized size = 0.95 \begin {gather*} \frac {\sqrt {\cos (a+b x)} \left (\frac {8 \cos (a+b x)}{3 b}+4 x \sin (a+b x)\right )}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )-\frac {x}{3 \sqrt {\cos \left (b x +a \right )}}\, dx\]
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*} \frac {\int \left (- \frac {x}{\sqrt {\cos {\left (a + b x \right )}}}\right )\, dx + \int 3 x \cos ^{\frac {3}{2}}{\left (a + b 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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\cos \left (a+b\,x\right )}^{3/2}-\frac {x}{3\,\sqrt {\cos \left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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