Optimal. Leaf size=47 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, 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 = {2716, 2720}
\begin {gather*} \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+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}}(a+b x)} \, dx &=-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+b x)}+\frac {1}{3} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 43, normalized size = 0.91 \begin {gather*} \frac {2 \left (F\left (\left .\frac {1}{4} (2 a-\pi +2 b x)\right |2\right )-\frac {\cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 88, normalized size = 1.87
method | result | size |
default | \(\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )^{\frac {3}{2}} \cos \left (b x +a \right ) b}\) | \(88\) |
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 = 115, normalized size = 2.45 \begin {gather*} \frac {\sqrt {-i} {\left (\sqrt {2} \cos \left (b x + a\right )^{2} - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + \sqrt {i} {\left (\sqrt {2} \cos \left (b x + a\right )^{2} - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, \cos \left (b x + a\right ) \sqrt {\sin \left (b x + a\right )}}{3 \, {\left (b \cos \left (b x + a\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 (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 = 0.61, size = 42, normalized size = 0.89 \begin {gather*} -\frac {\cos \left (a+b\,x\right )\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\sin \left (a+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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