Optimal. Leaf size=47 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A] time = 0.02, 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 = {2636, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
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 = 0.10, size = 43, normalized size = 0.91 \[ \frac {2 \left (F\left (\left .\frac {1}{4} (2 a+2 b x-\pi )\right |2\right )-\frac {\cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)}\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\sin \left (b x + a\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sin \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 88, normalized size = 1.87 \[ \frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )^{\frac {3}{2}} \cos \left (b x +a \right ) b} \]
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 \[ \int \frac {1}{\sin \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ -\frac {\cos \left (a+b\,x\right )\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\sin \left (a+b\,x\right )}^{3/2}} \]
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 \[ \int \frac {1}{\sin ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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