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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Maple [A]
time = 0.04, 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 [A]
time = 0.30, size = 18, normalized size = 1.64 \begin {gather*} \frac {\log \left (\sec \left (b x + a\right ) + \tan \left (b x + a\right )\right )}{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 = 2.38, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (8) = 16\).
time = 0.99, size = 36, normalized size = 3.27 \begin {gather*} \begin {cases} \frac {\log {\left (\tan {\left (a + b x \right )} + \sec {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \left (\tan {\left (a \right )} \sec {\left (a \right )} + \sec ^{2}{\left (a \right )}\right )}{\tan {\left (a \right )} + \sec {\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 = 0.44, size = 44, normalized size = 4.00 \begin {gather*} \frac {\log \left ({\left | \frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) - 2 \right |}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, 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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