Optimal. Leaf size=53 \[ \frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4303, 4292} \[ \frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4292
Rule 4303
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx &=-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {2}{3} \int \frac {\sin (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 43, normalized size = 0.81 \[ \frac {\sqrt {\sin (2 (a+b x))} \left (\frac {1}{4} \sec (a+b x)-\frac {1}{12} \cot (a+b x) \csc (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 74, normalized size = 1.40 \[ \frac {4 \, \cos \left (b x + a\right )^{3} + \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - 4 \, \cos \left (b x + a\right )}{12 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )}} \]
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 \[ \int \frac {\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 33.70, size = 194, normalized size = 3.66 \[ -\frac {\sqrt {-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (2 \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )-\left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{24 b \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}} \]
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 \[ \int \frac {\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.17, size = 104, normalized size = 1.96 \[ -\frac {2\,\sqrt {\sin \left (2\,a+2\,b\,x\right )}\,\left (3\,\cos \left (a+b\,x\right )-6\,\cos \left (3\,a+3\,b\,x\right )+4\,\cos \left (5\,a+5\,b\,x\right )-\cos \left (7\,a+7\,b\,x\right )\right )}{3\,b\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+4\,\cos \left (4\,a+4\,b\,x\right )-4\,\cos \left (6\,a+6\,b\,x\right )+\cos \left (8\,a+8\,b\,x\right )-5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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