Optimal. Leaf size=9 \[ b x+a \log (\sin (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3164, 3556}
\begin {gather*} a \log (\sin (x))+b x \end {gather*}
Antiderivative was successfully verified.
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Rule 3164
Rule 3556
Rubi steps
\begin {align*} \int \csc (x) (a \cos (x)+b \sin (x)) \, dx &=\int (b+a \cot (x)) \, dx\\ &=b x+a \int \cot (x) \, dx\\ &=b x+a \log (\sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} b x+a \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 10, normalized size = 1.11
method | result | size |
default | \(b x +a \ln \left (\sin \left (x \right )\right )\) | \(10\) |
risch | \(b x -i a x +a \ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(20\) |
norman | \(\frac {b x +b x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}+a \ln \left (\tan \left (\frac {x}{2}\right )\right )-a \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 9, normalized size = 1.00 \begin {gather*} b x + a \log \left (\sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.26, size = 11, normalized size = 1.22 \begin {gather*} b x + a \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.66, size = 8, normalized size = 0.89 \begin {gather*} a \log {\left (\sin {\left (x \right )} \right )} + b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs.
\(2 (9) = 18\).
time = 0.43, size = 24, normalized size = 2.67 \begin {gather*} b x - a \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.54, size = 54, normalized size = 6.00 \begin {gather*} a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )-a\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )-b\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,1{}\mathrm {i}+b\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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