Integrand size = 31, antiderivative size = 198 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{2} a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) x+\frac {b^3 (A b+4 a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \left (2 a^2 A+9 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d} \]
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Time = 0.63 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4110, 4179, 4161, 4132, 8, 4130, 3855} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^2 \left (2 a^2 A+9 a b B+9 A b^2\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (3 a^2 B+8 a A b-6 b^2 B\right ) \tan (c+d x)}{6 d}+\frac {1}{2} a x \left (a^3 B+4 a^2 A b+12 a b^2 B+8 A b^3\right )+\frac {b^3 (4 a B+A b) \text {arctanh}(\sin (c+d x))}{d}+\frac {a (a B+2 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d} \]
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Rule 8
Rule 3855
Rule 4110
Rule 4130
Rule 4132
Rule 4161
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (-3 a (2 A b+a B)-\left (2 a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)+b (a A-3 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {1}{6} \int \cos (c+d x) (a+b \sec (c+d x)) \left (-2 a \left (2 a^2 A+9 A b^2+9 a b B\right )-\left (8 a^2 A b+6 A b^3+3 a^3 B+18 a b^2 B\right ) \sec (c+d x)+b \left (8 a A b+3 a^2 B-6 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}-\frac {1}{6} \int \cos (c+d x) \left (-2 a^2 \left (2 a^2 A+9 A b^2+9 a b B\right )-3 a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) \sec (c+d x)-6 b^3 (A b+4 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}-\frac {1}{6} \int \cos (c+d x) \left (-2 a^2 \left (2 a^2 A+9 A b^2+9 a b B\right )-6 b^3 (A b+4 a B) \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right )\right ) \int 1 \, dx \\ & = \frac {1}{2} a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) x+\frac {a^2 \left (2 a^2 A+9 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}+\left (b^3 (A b+4 a B)\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) x+\frac {b^3 (A b+4 a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \left (2 a^2 A+9 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d} \\ \end{align*}
Time = 2.86 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.30 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {6 a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) (c+d x)-12 b^3 (A b+4 a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^3 (A b+4 a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {12 b^4 B \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {12 b^4 B \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 a^2 \left (3 a^2 A+24 A b^2+16 a b B\right ) \sin (c+d x)+3 a^3 (4 A b+a B) \sin (2 (c+d x))+a^4 A \sin (3 (c+d x))}{12 d} \]
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Time = 3.36 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{3} b \sin \left (d x +c \right )+6 A \,a^{2} b^{2} \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (d x +c \right )+4 A a \,b^{3} \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \tan \left (d x +c \right ) b^{4}}{d}\) | \(189\) |
default | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{3} b \sin \left (d x +c \right )+6 A \,a^{2} b^{2} \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (d x +c \right )+4 A a \,b^{3} \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \tan \left (d x +c \right ) b^{4}}{d}\) | \(189\) |
parallelrisch | \(\frac {-24 b^{3} \cos \left (d x +c \right ) \left (A b +4 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 b^{3} \cos \left (d x +c \right ) \left (A b +4 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (10 a^{4} A +72 A \,a^{2} b^{2}+48 B \,a^{3} b \right ) \sin \left (2 d x +2 c \right )+\left (12 A \,a^{3} b +3 B \,a^{4}\right ) \sin \left (3 d x +3 c \right )+a^{4} A \sin \left (4 d x +4 c \right )+48 \left (A \,a^{2} b +2 A \,b^{3}+\frac {1}{4} B \,a^{3}+3 B a \,b^{2}\right ) d x a \cos \left (d x +c \right )+12 \left (A \,a^{3} b +\frac {1}{4} B \,a^{4}+2 B \,b^{4}\right ) \sin \left (d x +c \right )}{24 d \cos \left (d x +c \right )}\) | \(212\) |
risch | \(2 A \,a^{3} b x +4 A a \,b^{3} x +\frac {a^{4} x B}{2}+6 B \,a^{2} b^{2} x +\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3} b}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A \,a^{3} b}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2} b^{2}}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A \,a^{3} b}{2 d}+\frac {2 i B \,b^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3} b}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2} b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{d}+\frac {a^{4} A \sin \left (3 d x +3 c \right )}{12 d}\) | \(365\) |
norman | \(\frac {\left (2 A \,a^{3} b +4 A a \,b^{3}+\frac {1}{2} B \,a^{4}+6 B \,a^{2} b^{2}\right ) x +\left (-6 A \,a^{3} b -12 A a \,b^{3}-\frac {3}{2} B \,a^{4}-18 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-6 A \,a^{3} b -12 A a \,b^{3}-\frac {3}{2} B \,a^{4}-18 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-2 A \,a^{3} b -4 A a \,b^{3}-\frac {1}{2} B \,a^{4}-6 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-2 A \,a^{3} b -4 A a \,b^{3}-\frac {1}{2} B \,a^{4}-6 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (2 A \,a^{3} b +4 A a \,b^{3}+\frac {1}{2} B \,a^{4}+6 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (6 A \,a^{3} b +12 A a \,b^{3}+\frac {3}{2} B \,a^{4}+18 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (6 A \,a^{3} b +12 A a \,b^{3}+\frac {3}{2} B \,a^{4}+18 B \,a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (2 a^{4} A -4 A \,a^{3} b +12 A \,a^{2} b^{2}-B \,a^{4}+8 B \,a^{3} b -2 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {\left (2 a^{4} A +4 A \,a^{3} b +12 A \,a^{2} b^{2}+B \,a^{4}+8 B \,a^{3} b +2 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (26 a^{4} A -60 A \,a^{3} b -36 A \,a^{2} b^{2}-15 B \,a^{4}-24 B \,a^{3} b +18 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {\left (26 a^{4} A +60 A \,a^{3} b -36 A \,a^{2} b^{2}+15 B \,a^{4}-24 B \,a^{3} b -18 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}-\frac {8 a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a^{2} \left (5 A \,a^{2}-12 A a b +18 A \,b^{2}-3 B \,a^{2}+12 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {4 a^{2} \left (5 A \,a^{2}+12 A a b +18 A \,b^{2}+3 B \,a^{2}+12 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {b^{3} \left (A b +4 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b^{3} \left (A b +4 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(781\) |
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Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (B a^{4} + 4 \, A a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{4} \cos \left (d x + c\right )^{3} + 6 \, B b^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (A a^{4} + 6 \, B a^{3} b + 9 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 72 \, {\left (d x + c\right )} B a^{2} b^{2} - 48 \, {\left (d x + c\right )} A a b^{3} - 24 \, B a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a^{3} b \sin \left (d x + c\right ) - 72 \, A a^{2} b^{2} \sin \left (d x + c\right ) - 12 \, B b^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.87 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {\frac {12 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 16.43 (sec) , antiderivative size = 2523, normalized size of antiderivative = 12.74 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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