Integrand size = 31, antiderivative size = 173 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{8} a^4 (35 A+48 B) x+\frac {a^4 B \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(35 A+32 B) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Time = 0.45 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {4102, 4081, 3855} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac {(35 A+32 B) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (35 A+48 B)+\frac {a^4 B \text {arctanh}(\sin (c+d x))}{d}+\frac {(7 A+4 B) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 d}+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
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Rule 3855
Rule 4081
Rule 4102
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 (a (7 A+4 B)+4 a B \sec (c+d x)) \, dx \\ & = \frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {1}{12} \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (35 A+32 B)+12 a^2 B \sec (c+d x)\right ) \, dx \\ & = \frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(35 A+32 B) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^3 (7 A+8 B)+24 a^3 B \sec (c+d x)\right ) \, dx \\ & = \frac {5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(35 A+32 B) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {1}{24} \int \left (-3 a^4 (35 A+48 B)-24 a^4 B \sec (c+d x)\right ) \, dx \\ & = \frac {1}{8} a^4 (35 A+48 B) x+\frac {5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(35 A+32 B) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 B\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} a^4 (35 A+48 B) x+\frac {a^4 B \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(7 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(35 A+32 B) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^4 \left (420 A d x+576 B d x-96 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 (28 A+27 B) \sin (c+d x)+24 (7 A+4 B) \sin (2 (c+d x))+32 A \sin (3 (c+d x))+8 B \sin (3 (c+d x))+3 A \sin (4 (c+d x))\right )}{96 d} \]
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Time = 2.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {\left (-32 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+32 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+8 \left (7 A +4 B \right ) \sin \left (2 d x +2 c \right )+\frac {8 \left (4 A +B \right ) \sin \left (3 d x +3 c \right )}{3}+A \sin \left (4 d x +4 c \right )+8 \left (28 A +27 B \right ) \sin \left (d x +c \right )+140 d \left (A +\frac {48 B}{35}\right ) x \right ) a^{4}}{32 d}\) | \(109\) |
derivativedivides | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (d x +c \right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 B \,a^{4} \sin \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(208\) |
default | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (d x +c \right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 B \,a^{4} \sin \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(208\) |
risch | \(\frac {35 a^{4} A x}{8}+6 a^{4} x B -\frac {7 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {7 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {a^{4} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} A \sin \left (3 d x +3 c \right )}{3 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{12 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{4}}{d}\) | \(224\) |
norman | \(\frac {\left (\frac {35}{8} a^{4} A +6 B \,a^{4}\right ) x +\left (-\frac {35}{2} a^{4} A -24 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {35}{2} a^{4} A -24 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {35}{8} a^{4} A +6 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {105}{4} a^{4} A +36 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {5 a^{4} \left (A -24 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}+\frac {25 a^{4} \left (5 A +8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {7 a^{4} \left (5 A +8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}-\frac {19 a^{4} \left (7 A +8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {5 a^{4} \left (7 A +8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}+\frac {3 a^{4} \left (31 A +24 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {5 a^{4} \left (103 A +136 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 d}-\frac {a^{4} \left (605 A +344 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {B \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(398\) |
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Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.68 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (35 \, A + 48 \, B\right )} a^{4} d x + 12 \, B a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, B a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (27 \, A + 16 \, B\right )} a^{4} \cos \left (d x + c\right ) + 160 \, {\left (A + B\right )} a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.18 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 96 \, {\left (d x + c\right )} A a^{4} + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 384 \, {\left (d x + c\right )} B a^{4} - 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 384 \, A a^{4} \sin \left (d x + c\right ) - 576 \, B a^{4} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.24 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {24 \, B a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, B a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (35 \, A a^{4} + 48 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 424 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 13.91 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.09 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {105\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+144\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+24\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+21\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+4\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+12\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )+B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+84\,A\,a^4\,\sin \left (c+d\,x\right )+81\,B\,a^4\,\sin \left (c+d\,x\right )}{12\,d} \]
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