Integrand size = 39, antiderivative size = 157 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
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Time = 0.55 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3120, 3057, 12, 3855} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {A \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rule 12
Rule 3057
Rule 3120
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(7 a A-a (3 A-3 B-4 C) \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (35 a^2 A-2 a^2 (10 A-3 B-4 C) \cos (c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (105 a^3 A-a^3 (55 A-6 B-8 C) \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int 105 a^4 A \sec (c+d x) \, dx}{105 a^8} \\ & = -\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {A \int \sec (c+d x) \, dx}{a^4} \\ & = \frac {A \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(334\) vs. \(2(157)=314\).
Time = 4.50 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.13 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \left (6720 A \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (70 (49 A-3 B-2 C) \sin \left (\frac {d x}{2}\right )-70 (31 A-2 C) \sin \left (c+\frac {d x}{2}\right )+2625 A \sin \left (c+\frac {3 d x}{2}\right )-126 B \sin \left (c+\frac {3 d x}{2}\right )-168 C \sin \left (c+\frac {3 d x}{2}\right )-735 A \sin \left (2 c+\frac {3 d x}{2}\right )+1015 A \sin \left (2 c+\frac {5 d x}{2}\right )-42 B \sin \left (2 c+\frac {5 d x}{2}\right )-56 C \sin \left (2 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {5 d x}{2}\right )+160 A \sin \left (3 c+\frac {7 d x}{2}\right )-6 B \sin \left (3 c+\frac {7 d x}{2}\right )-8 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )\right )}{210 a^4 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 (c+d x)))} \]
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Time = 1.99 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {-840 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+840 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (A -B +C \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (7 A -\frac {21 B}{5}+\frac {7 C}{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {77 A}{3}-7 B -\frac {7 C}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 A -7 B -7 C \right )}{840 a^{4} d}\) | \(123\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+8 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d \,a^{4}}\) | \(199\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+8 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d \,a^{4}}\) | \(199\) |
norman | \(\frac {-\frac {\left (A -B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {\left (15 A -B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (45 A -31 B +17 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}-\frac {\left (101 A -9 B -7 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}-\frac {\left (175 A -27 B -11 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}-\frac {\left (305 A -123 B +11 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{420 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{3}}+\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4} d}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4} d}\) | \(231\) |
risch | \(-\frac {2 i \left (105 A \,{\mathrm e}^{6 i \left (d x +c \right )}+735 A \,{\mathrm e}^{5 i \left (d x +c \right )}+2170 A \,{\mathrm e}^{4 i \left (d x +c \right )}-140 C \,{\mathrm e}^{4 i \left (d x +c \right )}+3430 A \,{\mathrm e}^{3 i \left (d x +c \right )}-210 B \,{\mathrm e}^{3 i \left (d x +c \right )}-140 C \,{\mathrm e}^{3 i \left (d x +c \right )}+2625 A \,{\mathrm e}^{2 i \left (d x +c \right )}-126 B \,{\mathrm e}^{2 i \left (d x +c \right )}-168 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1015 A \,{\mathrm e}^{i \left (d x +c \right )}-42 B \,{\mathrm e}^{i \left (d x +c \right )}-56 C \,{\mathrm e}^{i \left (d x +c \right )}+160 A -6 B -8 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}+\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{4} d}-\frac {A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{4} d}\) | \(233\) |
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Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left (A \cos \left (d x + c\right )^{4} + 4 \, A \cos \left (d x + c\right )^{3} + 6 \, A \cos \left (d x + c\right )^{2} + 4 \, A \cos \left (d x + c\right ) + A\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (A \cos \left (d x + c\right )^{4} + 4 \, A \cos \left (d x + c\right )^{3} + 6 \, A \cos \left (d x + c\right )^{2} + 4 \, A \cos \left (d x + c\right ) + A\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, {\left (80 \, A - 3 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (535 \, A - 24 \, B - 32 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (620 \, A - 39 \, B - 52 \, C\right )} \cos \left (d x + c\right ) + 260 \, A - 36 \, B - 13 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\int \frac {A \sec {\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (149) = 298\).
Time = 0.22 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.99 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - \frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} - \frac {3 \, B {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {840 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 1.24 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.27 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {2\,A\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B+C}{8\,a^4}+\frac {4\,A-2\,B}{8\,a^4}+\frac {4\,A+2\,B}{8\,a^4}+\frac {6\,A-2\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B+C}{24\,a^4}+\frac {4\,A-2\,B}{24\,a^4}+\frac {6\,A-2\,C}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B+C}{40\,a^4}+\frac {4\,A-2\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \]
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