Optimal. Leaf size=35 \[ (A b+a B) x+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b B \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3047, 3102,
2814, 3855} \begin {gather*} x (a B+A b)+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b B \sin (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2814
Rule 3047
Rule 3102
Rule 3855
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (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 (c+d x) \, dx\\ &=\frac {b B \sin (c+d x)}{d}+\int (a A+(A b+a B) \cos (c+d x)) \sec (c+d x) \, dx\\ &=(A b+a B) x+\frac {b B \sin (c+d x)}{d}+(a A) \int \sec (c+d x) \, dx\\ &=(A b+a B) x+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b B \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 1.31 \begin {gather*} A b x+a B x+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b B \cos (d x) \sin (c)}{d}+\frac {b B \cos (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 48, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \left (d x +c \right )+A b \left (d x +c \right )+B b \sin \left (d x +c \right )}{d}\) | \(48\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \left (d x +c \right )+A b \left (d x +c \right )+B b \sin \left (d x +c \right )}{d}\) | \(48\) |
risch | \(x A b +a B x -\frac {i B b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i B b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(83\) |
norman | \(\frac {\left (A b +a B \right ) x +\left (A b +a B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 A b +2 a B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 B b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 B b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 47, normalized size = 1.34 \begin {gather*} \frac {{\left (d x + c\right )} B a + {\left (d x + c\right )} A b + A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + B b \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 54, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (B a + A b\right )} d x + A a \log \left (\sin \left (d x + c\right ) + 1\right ) - A a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B b \sin \left (d x + c\right )}{2 \, d} \end {gather*}
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 \begin {gather*} \int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs.
\(2 (35) = 70\).
time = 0.53, size = 79, normalized size = 2.26 \begin {gather*} \frac {A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (B a + A b\right )} {\left (d x + c\right )} + \frac {2 \, B b \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 100, normalized size = 2.86 \begin {gather*} \frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,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\,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\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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