Optimal. Leaf size=24 \[ -\frac {\cos (a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {4291} \[ -\frac {\cos (a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 4291
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx &=-\frac {\cos (a+b x)}{b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.96 \[ -\frac {\cos (a+b x)}{b \sqrt {\sin (2 (a+b x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 39, normalized size = 1.62 \[ -\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + \sin \left (b x + a\right )}{2 \, b \sin \left (b x + a\right )} \]
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 \[ \int \frac {\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 10.74, size = 55902198, normalized size = 2329258.25 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (b x + a\right )}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 24, normalized size = 1.00 \[ -\frac {\sqrt {\sin \left (2\,a+2\,b\,x\right )}}{2\,b\,\sin \left (a+b\,x\right )} \]
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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