Optimal. Leaf size=24 \[ \frac {a \log (\sin (c+d x))}{d}+\frac {b \sin (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2721, 43} \[ \frac {a \log (\sin (c+d x))}{d}+\frac {b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2721
Rubi steps
\begin {align*} \int \cot (c+d x) (a+b \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+x}{x} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a}{x}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {a \log (\sin (c+d x))}{d}+\frac {b \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 1.79 \[ \frac {a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac {b \sin (c) \cos (d x)}{d}+\frac {b \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 24, normalized size = 1.00 \[ \frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 23, normalized size = 0.96 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 25, normalized size = 1.04 \[ \frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {b \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 22, normalized size = 0.92 \[ \frac {a \log \left (\sin \left (d x + c\right )\right ) + b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.56, size = 47, normalized size = 1.96 \[ \frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}+\frac {b\,\sin \left (c+d\,x\right )}{d} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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