Optimal. Leaf size=35 \[ \frac {(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a B \tan (c+d x)}{d}+b C x \]
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Rubi [A] time = 0.17, antiderivative size = 35, 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, 3021, 2735, 3770} \[ \frac {(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a B \tan (c+d x)}{d}+b C x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 3021
Rule 3029
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\int (a+b \cos (c+d x)) (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\int \left (a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a B \tan (c+d x)}{d}+\int (b B+a C+b C \cos (c+d x)) \sec (c+d x) \, dx\\ &=b C x+\frac {a B \tan (c+d x)}{d}-(-b B-a C) \int \sec (c+d x) \, dx\\ &=b C x+\frac {(b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a B \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.23 \[ \frac {a B \tan (c+d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+b C x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 85, normalized size = 2.43 \[ \frac {2 \, C b d x \cos \left (d x + c\right ) + {\left (C a + B b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (C a + B b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 84, normalized size = 2.40 \[ \frac {{\left (d x + c\right )} C b + {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 65, normalized size = 1.86 \[ b C x +\frac {a B \tan \left (d x +c \right )}{d}+\frac {B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 73, normalized size = 2.09 \[ \frac {2 \, {\left (d x + c\right )} C b + C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 114, normalized size = 3.26 \[ \frac {2\,C\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {B\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d} \]
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 \[ \int \left (B + C \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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