Optimal. Leaf size=16 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
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Rubi [A] time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3010, 2735, 3770} \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3010
Rule 3770
Rubi steps
\begin {align*} \int \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\int (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=C x+B \int \sec (c+d x) \, dx\\ &=C x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+C x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 36, normalized size = 2.25 \[ \frac {2 \, C d x + B \log \left (\sin \left (d x + c\right ) + 1\right ) - B \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.63, size = 43, normalized size = 2.69 \[ \frac {{\left (d x + c\right )} C + B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 30, normalized size = 1.88 \[ C x +\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 37, normalized size = 2.31 \[ \frac {2 \, {\left (d x + c\right )} C + B {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 57, normalized size = 3.56 \[ \frac {2\,B\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (B + C \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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