Optimal. Leaf size=42 \[ \frac {2 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, 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 = {2715, 2720}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
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 = 0.03, size = 36, normalized size = 0.86 \begin {gather*} \frac {2 \left (F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (a+b x)\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs.
\(2(62)=124\).
time = 0.03, size = 179, normalized size = 4.26
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\) | \(179\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 68, normalized size = 1.62 \begin {gather*} \frac {2 \, \sqrt {\cos \left (b x + a\right )} \sin \left (b x + a\right ) - i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{3 \, b} \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*} \int \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \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 = 0.08, size = 35, normalized size = 0.83 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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