Optimal. Leaf size=12 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {21, 3770} \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3770
Rubi steps
\begin {align*} \int \frac {(a B+b B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx &=B \int \sec (c+d x) \, dx\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 31, normalized size = 2.58 \[ \frac {B \log \left (\sin \left (d x + c\right ) + 1\right ) - B \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 47, normalized size = 3.92 \[ \frac {B \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - B \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 1.67 \[ \frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
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 = 0.49, size = 16, normalized size = 1.33 \[ \frac {2\,B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.81, size = 39, normalized size = 3.25 \[ \begin {cases} \frac {B \log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\relax (c )}\right ) \sec {\relax (c )}}{a + b \cos {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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