Optimal. Leaf size=17 \[ \frac {\sin (x)}{a}-\frac {\log (\sin (x)+1)}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac {\sin (x)}{a}-\frac {\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a+a \csc (x)} \, dx &=\int \frac {\cos (x) \sin (x)}{a+a \sin (x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (a+x)} \, dx,x,a \sin (x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+x} \, dx,x,a \sin (x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{a+x}\right ) \, dx,x,a \sin (x)\right )}{a^2}\\ &=-\frac {\log (1+\sin (x))}{a}+\frac {\sin (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\sin (x)}{a}-\frac {\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 15, normalized size = 0.88 \[ -\frac {\log \left (\sin \relax (x) + 1\right ) - \sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 17, normalized size = 1.00 \[ -\frac {\log \left (\sin \relax (x) + 1\right )}{a} + \frac {\sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 27, normalized size = 1.59 \[ \frac {1}{a \csc \relax (x )}+\frac {\ln \left (\csc \relax (x )\right )}{a}-\frac {\ln \left (1+\csc \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 17, normalized size = 1.00 \[ -\frac {\log \left (\sin \relax (x) + 1\right )}{a} + \frac {\sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (\sin \relax (x)+1\right )-\sin \relax (x)}{a} \]
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 \[ \frac {\int \frac {\cos {\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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