Integrand size = 29, antiderivative size = 86 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac {b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3069, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {1}{2} x \left (2 a^2 B+4 a A b+b^2 B\right )+\frac {b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac {b B \sin (c+d x) (a+b \cos (c+d x))}{2 d} \]
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Rule 2814
Rule 3069
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 A+\left (4 a A b+2 a^2 B+b^2 B\right ) \cos (c+d x)+b (2 A b+3 a B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac {b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 A+\left (4 a A b+2 a^2 B+b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) x+\frac {b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac {b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac {b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.40 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {2 \left (4 a A b+2 a^2 B+b^2 B\right ) (c+d x)-4 a^2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 a^2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 b (A b+2 a B) \sin (c+d x)+b^2 B \sin (2 (c+d x))}{4 d} \]
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Time = 1.53 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \left (d x +c \right )+2 A a b \left (d x +c \right )+2 B \sin \left (d x +c \right ) a b +A \sin \left (d x +c \right ) b^{2}+B \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(94\) |
default | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \left (d x +c \right )+2 A a b \left (d x +c \right )+2 B \sin \left (d x +c \right ) a b +A \sin \left (d x +c \right ) b^{2}+B \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(94\) |
parts | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,b^{2}+2 B a b \right ) \sin \left (d x +c \right )}{d}+\frac {\left (2 A a b +B \,a^{2}\right ) \left (d x +c \right )}{d}+\frac {B \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(95\) |
parallelrisch | \(\frac {-4 A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+B \sin \left (2 d x +2 c \right ) b^{2}+\left (4 A \,b^{2}+8 B a b \right ) \sin \left (d x +c \right )+8 x d \left (A a b +\frac {1}{2} B \,a^{2}+\frac {1}{4} B \,b^{2}\right )}{4 d}\) | \(97\) |
risch | \(2 x A a b +a^{2} B x +\frac {b^{2} B x}{2}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{2}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B a b}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{2}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B a b}{d}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{2}}{4 d}\) | \(156\) |
norman | \(\frac {\left (2 A a b +B \,a^{2}+\frac {1}{2} B \,b^{2}\right ) x +\left (2 A a b +B \,a^{2}+\frac {1}{2} B \,b^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A a b +3 B \,a^{2}+\frac {3}{2} B \,b^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A a b +3 B \,a^{2}+\frac {3}{2} B \,b^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (2 A b +4 B a -B b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \left (2 A b +4 B a +B b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 b \left (A b +2 B a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(251\) |
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Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} d x + {\left (B b^{2} \cos \left (d x + c\right ) + 4 \, B a b + 2 \, A b^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} B a^{2} + 8 \, {\left (d x + c\right )} A a b + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{2} + 4 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 8 \, B a b \sin \left (d x + c\right ) + 4 \, A b^{2} \sin \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.07 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {2 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (4 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 0.76 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.97 \[ \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx=\frac {A\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {2\,B\,a\,b\,\sin \left (c+d\,x\right )}{d}+\frac {4\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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