Optimal. Leaf size=17 \[ -\frac {\log (1+\sin (x))}{a}+\frac {\sin (x)}{a} \]
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Rubi [A]
time = 0.04, 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 = {3957, 2912, 12,
45} \begin {gather*} \frac {\sin (x)}{a}-\frac {\log (\sin (x)+1)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2912
Rule 3957
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 {\text {Subst}\left (\int \frac {x}{a (a+x)} \, dx,x,a \sin (x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {x}{a+x} \, dx,x,a \sin (x)\right )}{a^2}\\ &=\frac {\text {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 = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {\log (1+\sin (x))}{a}+\frac {\sin (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 23, normalized size = 1.35
method | result | size |
derivativedivides | \(-\frac {\ln \left (1+\csc \left (x \right )\right )-\frac {1}{\csc \left (x \right )}-\ln \left (\csc \left (x \right )\right )}{a}\) | \(23\) |
default | \(-\frac {\ln \left (1+\csc \left (x \right )\right )-\frac {1}{\csc \left (x \right )}-\ln \left (\csc \left (x \right )\right )}{a}\) | \(23\) |
risch | \(\frac {i x}{a}-\frac {i {\mathrm e}^{i x}}{2 a}+\frac {i {\mathrm e}^{-i x}}{2 a}-\frac {2 \ln \left (i+{\mathrm e}^{i x}\right )}{a}\) | \(45\) |
norman | \(\frac {\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{a}-\frac {2 \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 17, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\sin \left (x\right ) + 1\right )}{a} + \frac {\sin \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.61, size = 15, normalized size = 0.88 \begin {gather*} -\frac {\log \left (\sin \left (x\right ) + 1\right ) - \sin \left (x\right )}{a} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cos {\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 17, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\sin \left (x\right ) + 1\right )}{a} + \frac {\sin \left (x\right )}{a} \end {gather*}
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 \begin {gather*} -\frac {\ln \left (\sin \left (x\right )+1\right )-\sin \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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