Integrand size = 41, antiderivative size = 225 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Time = 0.67 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4171, 4102, 4081, 3855} \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (28 A+35 B+48 C)+\frac {a^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {(28 A+35 B+20 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac {a (4 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
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Rule 3855
Rule 4081
Rule 4102
Rule 4171
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (a (4 A+5 B)+5 a C \sec (c+d x)) \, dx}{5 a} \\ & = \frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (a^2 (28 A+35 B+20 C)+20 a^2 C \sec (c+d x)\right ) \, dx}{20 a} \\ & = \frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (5 a^3 (28 A+35 B+32 C)+60 a^3 C \sec (c+d x)\right ) \, dx}{60 a} \\ & = \frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^4 (28 A+35 B+40 C)+120 a^4 C \sec (c+d x)\right ) \, dx}{120 a} \\ & = \frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-15 a^5 (28 A+35 B+48 C)-120 a^5 C \sec (c+d x)\right ) \, dx}{120 a} \\ & = \frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \\ \end{align*}
Time = 3.83 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.81 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (1680 A d x+2100 B d x+2880 C d x-480 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 (49 A+56 B+54 C) \sin (c+d x)+120 (8 A+7 B+4 C) \sin (2 (c+d x))+290 A \sin (3 (c+d x))+160 B \sin (3 (c+d x))+40 C \sin (3 (c+d x))+60 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x))\right )}{480 d} \]
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Time = 0.51 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {29 \left (-\frac {48 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{29}+\frac {48 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{29}+\frac {12 \left (8 A +7 B +4 C \right ) \sin \left (2 d x +2 c \right )}{29}+\left (A +\frac {16 B}{29}+\frac {4 C}{29}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (2 A +\frac {B}{2}\right ) \sin \left (4 d x +4 c \right )}{29}+\frac {3 A \sin \left (5 d x +5 c \right )}{145}+\frac {6 \left (49 A +56 B +54 C \right ) \sin \left (d x +c \right )}{29}+\frac {168 \left (A +\frac {5 B}{4}+\frac {12 C}{7}\right ) x d}{29}\right ) a^{4}}{48 d}\) | \(139\) |
derivativedivides | \(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+2 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(340\) |
default | \(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+2 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(340\) |
risch | \(\frac {7 a^{4} A x}{2}+\frac {35 a^{4} x B}{8}+6 a^{4} x C -\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {49 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{8 d}+\frac {49 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{8 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{4} A \sin \left (4 d x +4 c \right )}{8 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {29 a^{4} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{12 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {7 \sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} C}{d}\) | \(341\) |
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Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.68 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (28 \, A + 35 \, B + 48 \, C\right )} a^{4} d x + 60 \, C a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (34 \, A + 20 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (28 \, A + 27 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (83 \, A + 100 \, B + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.48 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 480 \, {\left (d x + c\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1920 \, {\left (d x + c\right )} C a^{4} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \sin \left (d x + c\right ) + 1920 \, B a^{4} \sin \left (d x + c\right ) + 2880 \, C a^{4} \sin \left (d x + c\right )}{480 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.50 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (28 \, A a^{4} + 35 \, B a^{4} + 48 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (420 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 600 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1960 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3584 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3160 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3680 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1500 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1080 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
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Time = 17.13 (sec) , antiderivative size = 1151, normalized size of antiderivative = 5.12 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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