Optimal. Leaf size=17 \[ a x-\frac {b \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4042, 3770} \[ a x-\frac {b \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4042
Rubi steps
\begin {align*} \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx &=-\int (-a+b \sec (c+d x)) \, dx\\ &=a x-b \int \sec (c+d x) \, dx\\ &=a x-\frac {b \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ a x-\frac {b \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 36, normalized size = 2.12 \[ \frac {2 \, a d x - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 43, normalized size = 2.53 \[ \frac {{\left (d x + c\right )} a - b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 31, normalized size = 1.82 \[ a x -\frac {b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {c a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 57, normalized size = 3.35 \[ \frac {2\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.72, size = 41, normalized size = 2.41 \[ a x - b \left (\begin {cases} \frac {x \left (\tan {\relax (c )} \sec {\relax (c )} + \sec ^{2}{\relax (c )}\right )}{\tan {\relax (c )} + \sec {\relax (c )}} & \text {for}\: d = 0 \\\frac {\log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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