Optimal. Leaf size=11 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 11, 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}(\sin (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rubi steps
\begin {align*} \int \sec (a+b x) \, dx &=\frac {\tanh ^{-1}(\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(\sin (a+b x))}{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 = 2.37, size = 43, normalized size = 3.91 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\text {Log}\left [1+\text {Tan}\left [\frac {a}{2}+\frac {b x}{2}\right ]\right ]-\text {Log}\left [-1+\text {Tan}\left [\frac {a}{2}+\frac {b x}{2}\right ]\right ]}{b},b\text {!=}0\right \}\right \},\frac {x}{\text {Cos}\left [a\right ]}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 19, normalized size = 1.73
method | result | size |
derivativedivides | \(\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\) | \(19\) |
default | \(\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\) | \(19\) |
norman | \(\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}\) | \(35\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{b}\) | \(37\) |
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.
\(2 (11) = 22\).
time = 0.27, size = 26, normalized size = 2.36 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \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. 28 vs.
\(2 (11) = 22\).
time = 0.34, size = 28, normalized size = 2.55 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \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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Sympy [A]
time = 0.31, size = 34, normalized size = 3.09 \begin {gather*} \begin {cases} - \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} - 1 \right )}}{b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\cos {\left (a \right )}} & \text {otherwise} \end {cases} \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. 44 vs.
\(2 (11) = 22\).
time = 0.00, size = 48, normalized size = 4.36 \begin {gather*} \frac {2 \left (-\frac {\ln \left |\sin \left (a+b x\right )+\frac 1{\sin \left (a+b x\right )}-2\right |}{8}+\frac {\ln \left |\sin \left (a+b x\right )+\frac 1{\sin \left (a+b x\right )}+2\right |}{8}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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