Optimal. Leaf size=35 \[ x (a C+b B)+\frac {a B \sin (c+d x)}{d}+\frac {b C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {4072, 3996, 3770} \[ x (a C+b B)+\frac {a B \sin (c+d x)}{d}+\frac {b C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4072
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos (c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac {a B \sin (c+d x)}{d}-\int (-b B-a C-b C \sec (c+d x)) \, dx\\ &=(b B+a C) x+\frac {a B \sin (c+d x)}{d}+(b C) \int \sec (c+d x) \, dx\\ &=(b B+a C) x+\frac {b C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a B \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.31 \[ \frac {a B \sin (c) \cos (d x)}{d}+\frac {a B \cos (c) \sin (d x)}{d}+a C x+b B x+\frac {b C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 54, normalized size = 1.54 \[ \frac {2 \, {\left (C a + B b\right )} d x + C b \log \left (\sin \left (d x + c\right ) + 1\right ) - C b \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 79, normalized size = 2.26 \[ \frac {C b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - C b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (C a + B b\right )} {\left (d x + c\right )} + \frac {2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 56, normalized size = 1.60 \[ B x b +a C x +\frac {a B \sin \left (d x +c \right )}{d}+\frac {B b c}{d}+\frac {C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 58, normalized size = 1.66 \[ \frac {2 \, {\left (d x + c\right )} C a + 2 \, {\left (d x + c\right )} B b + C b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 100, normalized size = 2.86 \[ \frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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