Integrand size = 40, antiderivative size = 136 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(3 B-C) x}{a^3}+\frac {2 (36 B-11 C) \sin (c+d x)}{15 a^3 d}-\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 B-C) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4157, 4105, 3872, 2717, 8} \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {2 (36 B-11 C) \sin (c+d x)}{15 a^3 d}-\frac {(3 B-C) \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {x (3 B-C)}{a^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(B-C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 2717
Rule 3872
Rule 4105
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (c+d x) (B+C \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx \\ & = -\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) (a (6 B-C)-3 a (B-C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (a^2 (27 B-7 C)-2 a^2 (9 B-4 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 B-C) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \cos (c+d x) \left (2 a^3 (36 B-11 C)-15 a^3 (3 B-C) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 B-C) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(2 (36 B-11 C)) \int \cos (c+d x) \, dx}{15 a^3}-\frac {(3 B-C) \int 1 \, dx}{a^3} \\ & = -\frac {(3 B-C) x}{a^3}+\frac {2 (36 B-11 C) \sin (c+d x)}{15 a^3 d}-\frac {(B-C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 B-4 C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 B-C) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(136)=272\).
Time = 1.61 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-300 (3 B-C) d x \cos \left (\frac {d x}{2}\right )-300 (3 B-C) d x \cos \left (c+\frac {d x}{2}\right )-450 B d x \cos \left (c+\frac {3 d x}{2}\right )+150 C d x \cos \left (c+\frac {3 d x}{2}\right )-450 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+150 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-90 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+30 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-90 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+30 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+1755 B \sin \left (\frac {d x}{2}\right )-740 C \sin \left (\frac {d x}{2}\right )-1125 B \sin \left (c+\frac {d x}{2}\right )+540 C \sin \left (c+\frac {d x}{2}\right )+1215 B \sin \left (c+\frac {3 d x}{2}\right )-460 C \sin \left (c+\frac {3 d x}{2}\right )-225 B \sin \left (2 c+\frac {3 d x}{2}\right )+180 C \sin \left (2 c+\frac {3 d x}{2}\right )+363 B \sin \left (2 c+\frac {5 d x}{2}\right )-128 C \sin \left (2 c+\frac {5 d x}{2}\right )+75 B \sin \left (3 c+\frac {5 d x}{2}\right )+15 B \sin \left (3 c+\frac {7 d x}{2}\right )+15 B \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{120 a^3 d (1+\cos (c+d x))^3} \]
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Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {729 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {26 B}{81}-\frac {64 C}{729}\right ) \cos \left (2 d x +2 c \right )+\frac {5 B \cos \left (3 d x +3 c \right )}{243}+\left (B -\frac {68 C}{243}\right ) \cos \left (d x +c \right )+\frac {58 B}{81}-\frac {152 C}{729}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-720 x d \left (B -\frac {C}{3}\right )}{240 a^{3} d}\) | \(89\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(136\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(136\) |
risch | \(-\frac {3 B x}{a^{3}}+\frac {x C}{a^{3}}-\frac {i B \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i B \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {2 i \left (90 B \,{\mathrm e}^{4 i \left (d x +c \right )}-45 C \,{\mathrm e}^{4 i \left (d x +c \right )}+300 B \,{\mathrm e}^{3 i \left (d x +c \right )}-135 C \,{\mathrm e}^{3 i \left (d x +c \right )}+420 B \,{\mathrm e}^{2 i \left (d x +c \right )}-185 C \,{\mathrm e}^{2 i \left (d x +c \right )}+270 B \,{\mathrm e}^{i \left (d x +c \right )}-115 C \,{\mathrm e}^{i \left (d x +c \right )}+72 B -32 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(178\) |
norman | \(\frac {\frac {\left (3 B -C \right ) x}{a}+\frac {\left (3 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{20 a d}-\frac {\left (3 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {\left (3 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {\left (25 B -7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (27 B -17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 a d}-\frac {\left (45 B -17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {\left (111 B -41 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{30 a d}+\frac {\left (201 B -61 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\) | \(271\) |
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Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (3 \, B - C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (3 \, B - C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, B - C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (3 \, B - C\right )} d x - {\left (15 \, B \cos \left (d x + c\right )^{3} + {\left (117 \, B - 32 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (57 \, B - 17 \, C\right )} \cos \left (d x + c\right ) + 72 \, B - 22 \, C\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \, B {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - C {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {60 \, {\left (d x + c\right )} {\left (3 \, B - C\right )}}{a^{3}} - \frac {120 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 15.89 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,B}{2\,a^3}+\frac {3\,\left (B-C\right )}{4\,a^3}+\frac {4\,B-2\,C}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {B-C}{6\,a^3}+\frac {4\,B-2\,C}{12\,a^3}\right )}{d}-\frac {x\,\left (3\,B-C\right )}{a^3}+\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (B-C\right )}{20\,a^3\,d} \]
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