Integrand size = 39, antiderivative size = 192 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^2 (3 A b+a B) x+\frac {\left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {b \left (9 a b B-a^2 (6 A-8 C)+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 d}-\frac {b^2 (6 a A-3 b B-5 a C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \]
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Time = 0.41 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4179, 4141, 4133, 3855, 3852, 8} \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {b \tan (c+d x) \left (-\left (a^2 (6 A-8 C)\right )+9 a b B+b^2 (3 A+2 C)\right )}{3 d}+a^2 x (a B+3 A b)+\frac {\left (2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 \tan (c+d x) \sec (c+d x) (6 a A-5 a C-3 b B)}{6 d}-\frac {b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rule 4141
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^2 \left (3 A b+a B+(b B+a C) \sec (c+d x)-b (3 A-C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+b \sec (c+d x)) \left (3 a (3 A b+a B)+\left (3 A b^2+6 a b B+3 a^2 C+2 b^2 C\right ) \sec (c+d x)-b (6 a A-3 b B-5 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 (6 a A-3 b B-5 a C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^2 (3 A b+a B)+3 \left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \sec (c+d x)+2 b \left (9 a b B-a^2 (6 A-8 C)+b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = a^2 (3 A b+a B) x+\frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 (6 a A-3 b B-5 a C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{2} \left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \int \sec (c+d x) \, dx+\frac {1}{3} \left (b \left (9 a b B-a^2 (6 A-8 C)+b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx \\ & = a^2 (3 A b+a B) x+\frac {\left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 (6 a A-3 b B-5 a C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {\left (b \left (9 a b B-a^2 (6 A-8 C)+b^2 (3 A+2 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = a^2 (3 A b+a B) x+\frac {\left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {b \left (9 a b B-a^2 (6 A-8 C)+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 d}-\frac {b^2 (6 a A-3 b B-5 a C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(509\) vs. \(2(192)=384\).
Time = 9.47 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.65 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (12 a^2 (3 A b+a B) (c+d x)-6 \left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 (2 A+C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 (9 a C+b (3 B+C))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 b \left (3 A b^2+9 a b B+9 a^2 C+2 b^2 C\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {b^2 (9 a C+b (3 B+C))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 b \left (3 A b^2+9 a b B+9 a^2 C+2 b^2 C\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+12 a^3 A \sin (c+d x)\right )}{6 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \]
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Time = 1.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {a^{3} A \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{2} b \left (d x +c \right )+3 B \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \tan \left (d x +c \right ) a^{2} b +3 a A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \tan \left (d x +c \right ) a \,b^{2}+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{3}+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(235\) |
default | \(\frac {a^{3} A \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{2} b \left (d x +c \right )+3 B \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \tan \left (d x +c \right ) a^{2} b +3 a A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \tan \left (d x +c \right ) a \,b^{2}+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{3}+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(235\) |
parallelrisch | \(\frac {-6 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\frac {B \,b^{3}}{6}+a \left (A +\frac {C}{2}\right ) b^{2}+B \,a^{2} b +\frac {a^{3} C}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\frac {B \,b^{3}}{6}+a \left (A +\frac {C}{2}\right ) b^{2}+B \,a^{2} b +\frac {a^{3} C}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 d \,a^{2} \left (A b +\frac {a B}{3}\right ) x \cos \left (3 d x +3 c \right )+2 b \left (\left (A +\frac {2 C}{3}\right ) b^{2}+3 B a b +3 C \,a^{2}\right ) \sin \left (3 d x +3 c \right )+2 \left (a^{3} A +B \,b^{3}+3 C a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+a^{3} A \sin \left (4 d x +4 c \right )+18 d \,a^{2} \left (A b +\frac {a B}{3}\right ) x \cos \left (d x +c \right )+2 \sin \left (d x +c \right ) b \left (b^{2} \left (A +2 C \right )+3 B a b +3 C \,a^{2}\right )}{2 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(300\) |
risch | \(3 a^{2} A b x +a^{3} B x -\frac {i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i b \left (3 B \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+9 C a b \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}-18 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-36 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 B \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-9 C b a \,{\mathrm e}^{i \left (d x +c \right )}-6 A \,b^{2}-18 B a b -18 C \,a^{2}-4 C \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a A \,b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{2} b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{3}}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{2}}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a A \,b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{2} b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{3}}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{2}}{2 d}\) | \(484\) |
norman | \(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x +\left (-9 A \,a^{2} b -3 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-9 A \,a^{2} b -3 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (3 A \,a^{2} b +B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (6 A \,a^{2} b +2 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (6 A \,a^{2} b +2 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (2 a^{3} A -2 A \,b^{3}-6 B a \,b^{2}+B \,b^{3}-6 a^{2} b C +3 C a \,b^{2}-2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {\left (2 a^{3} A +2 A \,b^{3}+6 B a \,b^{2}+B \,b^{3}+6 a^{2} b C +3 C a \,b^{2}+2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (6 a^{3} A -B \,b^{3}-3 C a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {4 \left (6 a^{3} A -3 A \,b^{3}-9 B a \,b^{2}-9 a^{2} b C -C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {4 \left (6 a^{3} A +3 A \,b^{3}+9 B a \,b^{2}+9 a^{2} b C +C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (6 a A \,b^{2}+6 B \,a^{2} b +B \,b^{3}+2 a^{3} C +3 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (6 a A \,b^{2}+6 B \,a^{2} b +B \,b^{3}+2 a^{3} C +3 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(534\) |
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Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.17 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, {\left (2 \, A + C\right )} a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, {\left (2 \, A + C\right )} a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 2 \, C b^{3} + 2 \, {\left (9 \, C a^{2} b + 9 \, B a b^{2} + {\left (3 \, A + 2 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.46 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (d x + c\right )} B a^{3} + 36 \, {\left (d x + c\right )} A a^{2} b + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{3} - 9 \, C a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, B b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, A a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 36 \, C a^{2} b \tan \left (d x + c\right ) + 36 \, B a b^{2} \tan \left (d x + c\right ) + 12 \, A b^{3} \tan \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (185) = 370\).
Time = 0.38 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.28 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {12 \, A a^{3} \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} + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 6 \, A a b^{2} + 3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 6 \, A a b^{2} + 3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{3} \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 = 19.07 (sec) , antiderivative size = 2437, normalized size of antiderivative = 12.69 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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