Integrand size = 38, antiderivative size = 114 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {(3 a B+4 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \]
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Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3108, 3047, 3100, 2827, 3852, 3853, 3855} \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {(3 a B+4 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(a C+b B) \tan ^3(c+d x)}{3 d}+\frac {(a C+b B) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rule 2827
Rule 3047
Rule 3100
Rule 3108
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a+b \cos (c+d x)) (B+C \cos (c+d x)) \sec ^5(c+d x) \, dx \\ & = \int \left (a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx \\ & = \frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (4 (b B+a C)+(3 a B+4 b C) \cos (c+d x)) \sec ^4(c+d x) \, dx \\ & = \frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+(b B+a C) \int \sec ^4(c+d x) \, dx+\frac {1}{4} (3 a B+4 b C) \int \sec ^3(c+d x) \, dx \\ & = \frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 a B+4 b C) \int \sec (c+d x) \, dx-\frac {(b B+a C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {(3 a B+4 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 (3 a B+4 b C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (9 a B+12 b C+8 (b B+a C) (2+\cos (2 (c+d x))) \sec (c+d x)+6 a B \sec ^2(c+d x)\right ) \tan (c+d x)}{24 d} \]
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Time = 3.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04
method | result | size |
parts | \(-\frac {\left (B b +a C \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(119\) |
derivativedivides | \(\frac {-a C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(131\) |
default | \(\frac {-a C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(131\) |
parallelrisch | \(\frac {-18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (B a +\frac {4 C b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (B a +\frac {4 C b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \left (B b +a C \right ) \sin \left (2 d x +2 c \right )+3 \left (3 B a +4 C b \right ) \sin \left (3 d x +3 c \right )+8 \left (B b +a C \right ) \sin \left (4 d x +4 c \right )+3 \sin \left (d x +c \right ) \left (11 B a +4 C b \right )}{12 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(192\) |
risch | \(-\frac {i \left (9 B a \,{\mathrm e}^{7 i \left (d x +c \right )}+12 C b \,{\mathrm e}^{7 i \left (d x +c \right )}+33 B a \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C b \,{\mathrm e}^{5 i \left (d x +c \right )}-48 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 C a \,{\mathrm e}^{4 i \left (d x +c \right )}-33 B a \,{\mathrm e}^{3 i \left (d x +c \right )}-12 C b \,{\mathrm e}^{3 i \left (d x +c \right )}-64 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-64 a C \,{\mathrm e}^{2 i \left (d x +c \right )}-9 B a \,{\mathrm e}^{i \left (d x +c \right )}-12 C b \,{\mathrm e}^{i \left (d x +c \right )}-16 B b -16 a C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 B a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C b}{2 d}+\frac {3 B a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C b}{2 d}\) | \(266\) |
norman | \(\frac {\frac {\left (3 B a -8 B b -8 a C -36 C b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (3 B a +8 B b +8 a C -36 C b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (5 B a -8 B b -8 a C +4 C b \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (5 B a +8 B b +8 a C +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (27 B a -40 B b -40 a C -36 C b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (27 B a +40 B b +40 a C -36 C b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (39 B a -8 B b -8 a C +12 C b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (39 B a +8 B b +8 a C +12 C b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {\left (3 B a +4 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (3 B a +4 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(348\) |
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Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{2} + 6 \, B a + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, B a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (106) = 212\).
Time = 0.36 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.67 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 6.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.70 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\left (\frac {5\,B\,a}{4}-2\,B\,b-2\,C\,a+C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,B\,a}{4}+\frac {10\,B\,b}{3}+\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,B\,a}{4}-\frac {10\,B\,b}{3}-\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,B\,a}{4}+2\,B\,b+2\,C\,a+C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,B\,a}{4}+C\,b\right )}{d} \]
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