Optimal. Leaf size=32 \[ \frac {a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac {a C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {4072, 3914, 3767, 8, 3770} \[ \frac {a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac {a C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3914
Rule 4072
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int (a+a \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=a B x+(a C) \int \sec ^2(c+d x) \, dx+(a (B+C)) \int \sec (c+d x) \, dx\\ &=a B x+\frac {a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(a C) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a B x+\frac {a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.34 \[ \frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac {a C \tan (c+d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 79, normalized size = 2.47 \[ \frac {2 \, B a d x \cos \left (d x + c\right ) + {\left (B + C\right )} a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B + C\right )} a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.84, size = 84, normalized size = 2.62 \[ \frac {{\left (d x + c\right )} B a + {\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 65, normalized size = 2.03 \[ a B x +\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B a c}{d}+\frac {a C \tan \left (d x +c \right )}{d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 73, normalized size = 2.28 \[ \frac {2 \, {\left (d x + c\right )} B a + B a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.00, size = 100, normalized size = 3.12 \[ \frac {C\,a\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ a \left (\int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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