Optimal. Leaf size=41 \[ -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{3 b}-\frac {2 \cos (b x)}{3 b \sin ^{\frac {3}{2}}(b x)} \]
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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 = {2716, 2720}
\begin {gather*} -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{3 b}-\frac {2 \cos (b x)}{3 b \sin ^{\frac {3}{2}}(b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{\frac {5}{2}}(b x)} \, dx &=-\frac {2 \cos (b x)}{3 b \sin ^{\frac {3}{2}}(b x)}+\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)}{3 b \sin ^{\frac {3}{2}}(b x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 33, normalized size = 0.80 \begin {gather*} -\frac {2 \left (F\left (\left .\frac {1}{4} (\pi -2 b x)\right |2\right )+\frac {\cos (b x)}{\sin ^{\frac {3}{2}}(b x)}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 72, normalized size = 1.76
method | result | size |
default | \(\frac {\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x \right )-2 \left (\cos ^{2}\left (b x \right )\right )}{3 \sin \left (b x \right )^{\frac {3}{2}} \cos \left (b x \right ) b}\) | \(72\) |
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 = 0.09, size = 97, normalized size = 2.37 \begin {gather*} \frac {\sqrt {-i} {\left (\sqrt {2} \cos \left (b x\right )^{2} - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right ) + \sqrt {i} {\left (\sqrt {2} \cos \left (b x\right )^{2} - \sqrt {2}\right )} {\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 \, {\left (b \cos \left (b x\right )^{2} - b\right )}} \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 \frac {1}{\sin ^{\frac {5}{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 = 0.59, size = 34, normalized size = 0.83 \begin {gather*} -\frac {\cos \left (b\,x\right )\,{\left ({\sin \left (b\,x\right )}^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (b\,x\right )}^2\right )}{b\,{\sin \left (b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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