Optimal. Leaf size=41 \[ -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{3 b}-\frac {2 \cos (b x) \sqrt {\sin (b x)}}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2720}
\begin {gather*} -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{3 b}-\frac {2 \sqrt {\sin (b x)} \cos (b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rubi steps
\begin {align*} \int \sin ^{\frac {3}{2}}(b x) \, dx &=-\frac {2 \cos (b x) \sqrt {\sin (b x)}}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\sin (b x)}} \, dx\\ &=-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{3 b}-\frac {2 \cos (b x) \sqrt {\sin (b x)}}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 33, normalized size = 0.80 \begin {gather*} -\frac {2 \left (F\left (\left .\frac {1}{4} (\pi -2 b x)\right |2\right )+\cos (b x) \sqrt {\sin (b x)}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 72, normalized size = 1.76
method | result | size |
default | \(\frac {\frac {\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \left (\cos ^{2}\left (b x \right )\right ) \sin \left (b x \right )}{3}}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b}\) | \(72\) |
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 = 0.11, size = 60, normalized size = 1.46 \begin {gather*} \frac {\sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right ) + \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) - i \, \sin \left (b x\right )\right ) - 2 \, \cos \left (b x\right ) \sqrt {\sin \left (b x\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 \sin ^{\frac {3}{2}}{\left (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 = 0.40, size = 34, normalized size = 0.83 \begin {gather*} -\frac {\cos \left (b\,x\right )\,{\sin \left (b\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (b\,x\right )}^2\right )}{b\,{\left ({\sin \left (b\,x\right )}^2\right )}^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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