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 = {3770} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3770
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 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 28, normalized size = 2.55 \[ \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.39, size = 28, normalized size = 2.55 \[ \frac {\log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 1.73 \[ \frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 26, normalized size = 2.36 \[ \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{2 \, b} \]
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 \[ \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 34, normalized size = 3.09 \[ \begin {cases} - \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} - 1 \right )}}{b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\cos {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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