Optimal. Leaf size=42 \[ \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2635, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.86 \[ \frac {2 \left (F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sin (a+b x) \sqrt {\cos (a+b x)}\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int \cos \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 179, normalized size = 4.26 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]
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 \[ \int \cos \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 35, normalized size = 0.83 \[ \frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{3\,b}+\frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\sin \left (a+b\,x\right )}{3\,b} \]
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 \[ \int \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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