Optimal. Leaf size=35 \[ \frac {(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}+b B x \]
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Rubi [A] time = 0.11, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2968, 3021, 2735, 3770} \[ \frac {(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}+b B x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 3021
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx &=\int \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a A \tan (c+d x)}{d}+\int (A b+a B+b B \cos (c+d x)) \sec (c+d x) \, dx\\ &=b B x+\frac {a A \tan (c+d x)}{d}-(-A b-a B) \int \sec (c+d x) \, dx\\ &=b B x+\frac {(A b+a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.23 \[ \frac {a A \tan (c+d x)}{d}+\frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A b \tanh ^{-1}(\sin (c+d x))}{d}+b B x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 85, normalized size = 2.43 \[ \frac {2 \, B b d x \cos \left (d x + c\right ) + {\left (B a + A b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B a + A b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 84, normalized size = 2.40 \[ \frac {{\left (d x + c\right )} B b + {\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 65, normalized size = 1.86 \[ b B x +\frac {a A \tan \left (d x +c \right )}{d}+\frac {A b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 73, normalized size = 2.09 \[ \frac {2 \, {\left (d x + c\right )} B b + B a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + A b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 114, normalized size = 3.26 \[ \frac {2\,B\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d} \]
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 \[ \int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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