Optimal. Leaf size=42 \[ \frac {a A \sin (c+d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac {b C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4077, 4047, 8, 4045, 3770} \[ \frac {a A \sin (c+d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac {b C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4045
Rule 4047
Rule 4077
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {b C \tan (c+d x)}{d}+\int \cos (c+d x) \left (a A+A b \sec (c+d x)+a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b C \tan (c+d x)}{d}+(A b) \int 1 \, dx+\int \cos (c+d x) \left (a A+a C \sec ^2(c+d x)\right ) \, dx\\ &=A b x+\frac {a A \sin (c+d x)}{d}+\frac {b C \tan (c+d x)}{d}+(a C) \int \sec (c+d x) \, dx\\ &=A b x+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x)}{d}+\frac {b C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.29 \[ \frac {a A \sin (c) \cos (d x)}{d}+\frac {a A \cos (c) \sin (d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac {b C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 86, normalized size = 2.05 \[ \frac {2 \, A b d x \cos \left (d x + c\right ) + C a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - C a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a \cos \left (d x + c\right ) + C b\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 119, normalized size = 2.83 \[ \frac {{\left (d x + c\right )} A b + C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 57, normalized size = 1.36 \[ A b x +\frac {a A \sin \left (d x +c \right )}{d}+\frac {A b c}{d}+\frac {b C \tan \left (d x +c \right )}{d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 59, normalized size = 1.40 \[ \frac {2 \, {\left (d x + c\right )} A b + C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \sin \left (d x + c\right ) + 2 \, C b \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 91, normalized size = 2.17 \[ \frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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 \[ \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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