Integrand size = 39, antiderivative size = 137 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {(3 a A+4 b B+4 a C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(2 A b+2 a B+3 b C) \tan (c+d x)}{3 d}+\frac {(3 a A+4 b B+4 a C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3110, 3100, 2827, 3853, 3855, 3852, 8} \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {(3 a A+4 a C+4 b B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) (2 a B+2 A b+3 b C)}{3 d}+\frac {\tan (c+d x) \sec (c+d x) (3 a A+4 a C+4 b B)}{8 d}+\frac {(a B+A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a A \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rule 8
Rule 2827
Rule 3100
Rule 3110
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{4} \int \left (-4 (A b+a B)-(3 a A+4 b B+4 a C) \cos (c+d x)-4 b C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {(A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{12} \int (-3 (3 a A+4 b B+4 a C)-4 (2 A b+2 a B+3 b C) \cos (c+d x)) \sec ^3(c+d x) \, dx \\ & = \frac {(A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{4} (-3 a A-4 b B-4 a C) \int \sec ^3(c+d x) \, dx-\frac {1}{3} (-2 A b-2 a B-3 b C) \int \sec ^2(c+d x) \, dx \\ & = \frac {(3 a A+4 b B+4 a C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} (-3 a A-4 b B-4 a C) \int \sec (c+d x) \, dx-\frac {(2 A b+2 a B+3 b C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = \frac {(3 a A+4 b B+4 a C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(2 A b+2 a B+3 b C) \tan (c+d x)}{3 d}+\frac {(3 a A+4 b B+4 a C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {a A \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.73 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 (3 a A+4 b B+4 a C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (24 (A b+a B+b C)+3 (3 a A+4 b B+4 a C) \sec (c+d x)+6 a A \sec ^3(c+d x)+8 (A b+a B) \tan ^2(c+d x)\right )}{24 d} \]
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Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99
method | result | size |
parts | \(\frac {a A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (A b +B a \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B b +C a \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C b \tan \left (d x +c \right )}{d}\) | \(136\) |
derivativedivides | \(\frac {a A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b}{d}\) | \(174\) |
default | \(\frac {a A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b}{d}\) | \(174\) |
parallelrisch | \(\frac {-9 \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \left (\left (A +\frac {4 C}{3}\right ) a +\frac {4 B b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \left (\left (A +\frac {4 C}{3}\right ) a +\frac {4 B b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+64 \left (\left (A +\frac {3 C}{4}\right ) b +B a \right ) \sin \left (2 d x +2 c \right )+6 \left (\left (3 A +4 C \right ) a +4 B b \right ) \sin \left (3 d x +3 c \right )+16 \left (B a +\left (A +\frac {3 C}{2}\right ) b \right ) \sin \left (4 d x +4 c \right )+66 \sin \left (d x +c \right ) \left (\left (A +\frac {4 C}{11}\right ) a +\frac {4 B b}{11}\right )}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(216\) |
norman | \(\frac {\frac {\left (7 a A -4 B b -4 C a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (27 a A -16 A b -16 B a +12 B b +12 C a \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {\left (27 a A +16 A b +16 B a +12 B b +12 C a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {\left (5 a A -8 A b -8 B a +4 B b +4 C a -8 C b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (5 a A +8 A b +8 B a +4 B b +4 C a +8 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (81 a A -8 A b -8 B a -12 B b -12 C a -72 C b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (81 a A +8 A b +8 B a -12 B b -12 C a +72 C b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {\left (3 a A +4 B b +4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (3 a A +4 B b +4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(358\) |
risch | \(-\frac {i \left (9 A a \,{\mathrm e}^{7 i \left (d x +c \right )}+12 B b \,{\mathrm e}^{7 i \left (d x +c \right )}+12 C a \,{\mathrm e}^{7 i \left (d x +c \right )}-24 C b \,{\mathrm e}^{6 i \left (d x +c \right )}+33 A a \,{\mathrm e}^{5 i \left (d x +c \right )}+12 B b \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C a \,{\mathrm e}^{5 i \left (d x +c \right )}-48 A b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 B a \,{\mathrm e}^{4 i \left (d x +c \right )}-72 C b \,{\mathrm e}^{4 i \left (d x +c \right )}-33 A a \,{\mathrm e}^{3 i \left (d x +c \right )}-12 B b \,{\mathrm e}^{3 i \left (d x +c \right )}-12 C a \,{\mathrm e}^{3 i \left (d x +c \right )}-64 A b \,{\mathrm e}^{2 i \left (d x +c \right )}-64 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-72 C b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 a A \,{\mathrm e}^{i \left (d x +c \right )}-12 B b \,{\mathrm e}^{i \left (d x +c \right )}-12 C a \,{\mathrm e}^{i \left (d x +c \right )}-16 A b -16 B a -24 C b \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C a}{2 d}+\frac {3 a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C a}{2 d}\) | \(401\) |
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a + 4 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a + 4 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (2 \, B a + {\left (2 \, A + 3 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a + 4 \, B b\right )} \cos \left (d x + c\right )^{2} + 6 \, A a + 8 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.59 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b - 3 \, A a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (127) = 254\).
Time = 0.33 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.12 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 \, {\left (3 \, A a + 4 \, C a + 4 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a + 4 \, C a + 4 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, 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 = 5.93 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.87 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,A\,a}{8}+\frac {B\,b}{2}+\frac {C\,a}{2}\right )}{\frac {3\,A\,a}{2}+2\,B\,b+2\,C\,a}\right )\,\left (\frac {3\,A\,a}{4}+B\,b+C\,a\right )}{d}+\frac {\left (\frac {5\,A\,a}{4}-2\,A\,b-2\,B\,a+B\,b+C\,a-2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,A\,a}{4}+\frac {10\,A\,b}{3}+\frac {10\,B\,a}{3}-B\,b-C\,a+6\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,A\,a}{4}-\frac {10\,A\,b}{3}-\frac {10\,B\,a}{3}-B\,b-C\,a-6\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A\,a}{4}+2\,A\,b+2\,B\,a+B\,b+C\,a+2\,C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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