Optimal. Leaf size=12 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855}
\begin {gather*} -\frac {\tanh ^{-1}(\cos (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rubi steps
\begin {align*} \int \csc (a+b x) \, dx &=-\frac {\tanh ^{-1}(\cos (a+b x))}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
time = 0.01, size = 38, normalized size = 3.17 \begin {gather*} -\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.96, size = 26, normalized size = 2.17 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\text {Log}\left [\text {Tan}\left [\frac {a}{2}+\frac {b x}{2}\right ]\right ]}{b},b\text {!=}0\right \}\right \},\frac {x}{\text {Sin}\left [a\right ]}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 21, normalized size = 1.75
method | result | size |
norman | \(\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\) | \(15\) |
derivativedivides | \(\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b}\) | \(21\) |
default | \(\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b}\) | \(21\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\).
time = 0.26, size = 26, normalized size = 2.17 \begin {gather*} -\frac {\log \left (\cos \left (b x + a\right ) + 1\right ) - \log \left (\cos \left (b x + a\right ) - 1\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (12) = 24\).
time = 0.34, size = 30, normalized size = 2.50 \begin {gather*} -\frac {\log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 17, normalized size = 1.42 \begin {gather*} \begin {cases} \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sin {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 17, normalized size = 1.42 \begin {gather*} \frac {2 \ln \left |\tan \left (\frac {b x+a}{2}\right )\right |}{b\cdot 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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