Integrand size = 33, antiderivative size = 219 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d} \]
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Time = 0.68 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4180, 4179, 4141, 4133, 3855, 3852, 8} \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a b \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {1}{2} a^2 x \left (a^2 (A+2 C)+12 A b^2\right )-\frac {a b^3 (9 A-4 C) \tan (c+d x) \sec (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^2}{6 d}+\frac {2 A b \sin (c+d x) (a+b \sec (c+d x))^3}{d}+\frac {A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^4}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rule 4141
Rule 4179
Rule 4180
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (A+2 C) \sec (c+d x)-b (3 A-2 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}+\frac {1}{2} \int (a+b \sec (c+d x))^2 \left (12 A b^2+a^2 (A+2 C)-2 a b (A-2 C) \sec (c+d x)-b^2 (15 A-2 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+b \sec (c+d x)) \left (3 a \left (12 A b^2+a^2 (A+2 C)\right )-b \left (3 a^2 (A-6 C)-2 b^2 (3 A+2 C)\right ) \sec (c+d x)-4 a b^2 (9 A-4 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{12} \int \left (6 a^2 \left (12 A b^2+a^2 (A+2 C)\right )+24 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \sec (c+d x)-2 b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\left (2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right )\right ) \int \sec (c+d x) \, dx-\frac {1}{6} \left (b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {\left (b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = \frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d} \\ \end{align*}
Time = 7.98 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.90 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {6 a^2 \left (12 A b^2+a^2 (A+2 C)\right ) (c+d x)-24 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^3 (12 a+b) C}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 b^2 \left (3 A b^2+2 \left (9 a^2+b^2\right ) C\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {b^3 (12 a+b) C}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 b^2 \left (3 A b^2+2 \left (9 a^2+b^2\right ) C\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+48 a^3 A b \sin (c+d x)+3 a^4 A \sin (2 (c+d x))}{12 d} \]
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Time = 1.13 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (d x +c \right )+4 A \,a^{3} b \sin \left (d x +c \right )+4 a^{3} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{2} b^{2} \left (d x +c \right )+6 C \tan \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{4}-C \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(200\) |
default | \(\frac {a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (d x +c \right )+4 A \,a^{3} b \sin \left (d x +c \right )+4 a^{3} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{2} b^{2} \left (d x +c \right )+6 C \tan \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{4}-C \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(200\) |
parallelrisch | \(\frac {-96 \left (C \,a^{2}+\left (A +\frac {C}{2}\right ) b^{2}\right ) a b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+96 \left (C \,a^{2}+\left (A +\frac {C}{2}\right ) b^{2}\right ) a b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 \left (12 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) a^{2} x d \cos \left (3 d x +3 c \right )+\left (3 a^{4} A +48 C \,a^{2} b^{2}+8 \left (A +\frac {2 C}{3}\right ) b^{4}\right ) \sin \left (3 d x +3 c \right )+32 \left (A \,a^{3} b +C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+16 A \,a^{3} b \sin \left (4 d x +4 c \right )+a^{4} A \sin \left (5 d x +5 c \right )+12 \left (12 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) a^{2} x d \cos \left (d x +c \right )+2 \left (a^{4} A +24 C \,a^{2} b^{2}+4 b^{4} \left (A +2 C \right )\right ) \sin \left (d x +c \right )}{8 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(309\) |
risch | \(\frac {a^{4} A x}{2}+6 A \,a^{2} b^{2} x +a^{4} x C -\frac {i a^{4} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 i A \,a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {2 i A \,a^{3} b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i a^{4} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 i b^{2} \left (6 C a b \,{\mathrm e}^{5 i \left (d x +c \right )}-3 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C b a \,{\mathrm e}^{i \left (d x +c \right )}-3 A \,b^{2}-18 C \,a^{2}-2 C \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(385\) |
norman | \(\frac {\left (-\frac {1}{2} a^{4} A -6 A \,a^{2} b^{2}-a^{4} C \right ) x +\left (-\frac {5}{2} a^{4} A -30 A \,a^{2} b^{2}-5 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {3}{2} a^{4} A -18 A \,a^{2} b^{2}-3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-\frac {1}{2} a^{4} A -6 A \,a^{2} b^{2}-a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {1}{2} a^{4} A +6 A \,a^{2} b^{2}+a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {1}{2} a^{4} A +6 A \,a^{2} b^{2}+a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {3}{2} a^{4} A +18 A \,a^{2} b^{2}+3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {5}{2} a^{4} A +30 A \,a^{2} b^{2}+5 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {4 \left (15 a^{4} A -6 A \,b^{4}-36 C \,a^{2} b^{2}-2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {2 \left (9 a^{4} A -48 A \,a^{3} b +6 A \,b^{4}+36 C \,a^{2} b^{2}+2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {2 \left (9 a^{4} A +48 A \,a^{3} b +6 A \,b^{4}+36 C \,a^{2} b^{2}+2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {\left (a^{4} A -8 A \,a^{3} b +2 A \,b^{4}+12 C \,a^{2} b^{2}-4 C a \,b^{3}+2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}-\frac {\left (a^{4} A +8 A \,a^{3} b +2 A \,b^{4}+12 C \,a^{2} b^{2}+4 C a \,b^{3}+2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (15 a^{4} A -40 A \,a^{3} b -2 A \,b^{4}-12 C \,a^{2} b^{2}+12 C a \,b^{3}-2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {\left (15 a^{4} A +40 A \,a^{3} b -2 A \,b^{4}-12 C \,a^{2} b^{2}-12 C a \,b^{3}-2 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {2 a b \left (2 A \,b^{2}+2 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a b \left (2 A \,b^{2}+2 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(749\) |
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Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.95 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 12 \, A a^{2} b^{2}\right )} d x \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, C a^{3} b + {\left (2 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6 \, {\left (2 \, C a^{3} b + {\left (2 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (3 \, A a^{4} \cos \left (d x + c\right )^{4} + 24 \, A a^{3} b \cos \left (d x + c\right )^{3} + 12 \, C a b^{3} \cos \left (d x + c\right ) + 2 \, C b^{4} + 2 \, {\left (18 \, C a^{2} b^{2} + {\left (3 \, A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 12 \, {\left (d x + c\right )} C a^{4} + 72 \, {\left (d x + c\right )} A a^{2} b^{2} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{4} - 12 \, C a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{3} b \sin \left (d x + c\right ) + 72 \, C a^{2} b^{2} \tan \left (d x + c\right ) + 12 \, A b^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.81 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (A a^{4} + 2 \, C a^{4} + 12 \, A a^{2} b^{2}\right )} {\left (d x + c\right )} + 12 \, {\left (2 \, C a^{3} b + 2 \, A a b^{3} + C a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, {\left (2 \, C a^{3} b + 2 \, A a b^{3} + C a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {4 \, {\left (18 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 18.42 (sec) , antiderivative size = 2664, normalized size of antiderivative = 12.16 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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