Integrand size = 33, antiderivative size = 182 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \sin (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d} \]
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Time = 0.60 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4180, 4179, 4159, 4132, 8, 4130, 3855} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (a^2 (4 A+6 C)+A b^2\right ) \sin (c+d x)}{2 d}+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac {b^3 C \text {arctanh}(\sin (c+d x))}{d} \]
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Rule 8
Rule 3855
Rule 4130
Rule 4132
Rule 4159
Rule 4179
Rule 4180
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (3 A b+a (3 A+4 C) \sec (c+d x)+4 b C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (3 \left (2 A b^2+a^2 (3 A+4 C)\right )+3 a b (5 A+8 C) \sec (c+d x)+12 b^2 C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-12 b \left (A b^2+a^2 (4 A+6 C)\right )-3 a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \sec (c+d x)-24 b^3 C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-12 b \left (A b^2+a^2 (4 A+6 C)\right )-24 b^3 C \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right )\right ) \int 1 \, dx \\ & = \frac {1}{8} a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \sin (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\left (b^3 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \sin (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d} \\ \end{align*}
Time = 1.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) (c+d x)-32 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b \left (4 A b^2+3 a^2 (3 A+4 C)\right ) \sin (c+d x)+8 a \left (3 A b^2+a^2 (A+C)\right ) \sin (2 (c+d x))+8 a^2 A b \sin (3 (c+d x))+a^3 A \sin (4 (c+d x))}{32 d} \]
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Time = 0.65 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {-32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3}+32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}+8 a \left (\left (A +C \right ) a^{2}+3 A \,b^{2}\right ) \sin \left (2 d x +2 c \right )+8 A \,a^{2} b \sin \left (3 d x +3 c \right )+a^{3} A \sin \left (4 d x +4 c \right )+72 b \left (\left (A +\frac {4 C}{3}\right ) a^{2}+\frac {4 A \,b^{2}}{9}\right ) \sin \left (d x +c \right )+12 a x \left (\left (A +\frac {4 C}{3}\right ) a^{2}+4 b^{2} \left (A +2 C \right )\right ) d}{32 d}\) | \(149\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C \,a^{2} b \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \sin \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(177\) |
default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C \,a^{2} b \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \sin \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(177\) |
risch | \(\frac {3 a^{3} A x}{8}+\frac {3 A a \,b^{2} x}{2}+\frac {a^{3} x C}{2}+3 C a \,b^{2} x +\frac {9 i A \,a^{2} b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {9 i A \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b C}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{3}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{3}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b C}{2 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{3} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {A \,a^{2} b \sin \left (3 d x +3 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}\) | \(285\) |
norman | \(\frac {\left (\frac {3}{8} a^{3} A +\frac {3}{2} a A \,b^{2}+\frac {1}{2} a^{3} C +3 C a \,b^{2}\right ) x +\left (-\frac {3}{2} a^{3} A -6 a A \,b^{2}-2 a^{3} C -12 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {3}{2} a^{3} A -6 a A \,b^{2}-2 a^{3} C -12 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {3}{8} a^{3} A +\frac {3}{2} a A \,b^{2}+\frac {1}{2} a^{3} C +3 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {9}{4} a^{3} A +9 a A \,b^{2}+3 a^{3} C +18 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (5 a^{3} A -24 A \,a^{2} b +12 a A \,b^{2}-8 A \,b^{3}+4 a^{3} C -24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}+\frac {\left (5 a^{3} A +24 A \,a^{2} b +12 a A \,b^{2}+8 A \,b^{3}+4 a^{3} C +24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (23 a^{3} A -56 A \,a^{2} b +36 a A \,b^{2}-8 A \,b^{3}+12 a^{3} C -24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}-\frac {\left (23 a^{3} A +56 A \,a^{2} b +36 a A \,b^{2}+8 A \,b^{3}+12 a^{3} C +24 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}-\frac {\left (45 a^{3} A -24 A \,a^{2} b +12 a A \,b^{2}+24 A \,b^{3}+4 a^{3} C +72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {\left (45 a^{3} A +24 A \,a^{2} b +12 a A \,b^{2}-24 A \,b^{3}+4 a^{3} C -72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {\left (55 a^{3} A -8 A \,a^{2} b -60 a A \,b^{2}-24 A \,b^{3}-20 a^{3} C -72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (55 a^{3} A +8 A \,a^{2} b -60 a A \,b^{2}+24 A \,b^{3}-20 a^{3} C +72 a^{2} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(710\) |
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Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.80 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, C b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, C b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, {\left (A + 2 \, C\right )} a b^{2}\right )} d x + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 8 \, A a^{2} b \cos \left (d x + c\right )^{2} + 8 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 8 \, A b^{3} + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 96 \, {\left (d x + c\right )} C a b^{2} + 16 \, C b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, C a^{2} b \sin \left (d x + c\right ) + 32 \, A b^{3} \sin \left (d x + c\right )}{32 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (173) = 346\).
Time = 0.36 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.76 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (3 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 24 \, C a b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A 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 )}^{4}}}{8 \, d} \]
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Time = 17.21 (sec) , antiderivative size = 2008, normalized size of antiderivative = 11.03 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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