Optimal. Leaf size=32 \[ \frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac {a C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.15, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3029, 2968, 3023, 2735, 3770} \[ \frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac {a C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 3023
Rule 3029
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\int (a+a \cos (c+d x)) (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=\int \left (a B+(a B+a C) \cos (c+d x)+a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a C \sin (c+d x)}{d}+\int (a B+a (B+C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=a (B+C) x+\frac {a C \sin (c+d x)}{d}+(a B) \int \sec (c+d x) \, dx\\ &=a (B+C) x+\frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a C \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 1.44 \[ \frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac {a C \sin (c) \cos (d x)}{d}+\frac {a C \cos (c) \sin (d x)}{d}+a C x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 51, normalized size = 1.59 \[ \frac {2 \, {\left (B + C\right )} a d x + B a \log \left (\sin \left (d x + c\right ) + 1\right ) - B a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 79, normalized size = 2.47 \[ \frac {B a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - B a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (B a + C a\right )} {\left (d x + c\right )} + \frac {2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 56, normalized size = 1.75 \[ a B x +a C x +\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B a c}{d}+\frac {a C \sin \left (d x +c \right )}{d}+\frac {C a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 58, normalized size = 1.81 \[ \frac {2 \, {\left (d x + c\right )} B a + 2 \, {\left (d x + c\right )} C a + B a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 100, normalized size = 3.12 \[ \frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ a \left (\int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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