Integrand size = 40, antiderivative size = 98 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {(2 B-C) x}{a^2}+\frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
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Time = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4157, 4105, 3872, 2717, 8} \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac {x (2 B-C)}{a^2}-\frac {(B-C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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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))^2} \, dx \\ & = -\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) (a (4 B-C)-2 a (B-C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \cos (c+d x) \left (2 a^2 (5 B-2 C)-3 a^2 (2 B-C) \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(2 (5 B-2 C)) \int \cos (c+d x) \, dx}{3 a^2}-\frac {(2 B-C) \int 1 \, dx}{a^2} \\ & = -\frac {(2 B-C) x}{a^2}+\frac {2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac {(2 B-C) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(B-C) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(98)=196\).
Time = 0.88 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-18 (2 B-C) d x \cos \left (\frac {d x}{2}\right )-18 (2 B-C) d x \cos \left (c+\frac {d x}{2}\right )-12 B d x \cos \left (c+\frac {3 d x}{2}\right )+6 C d x \cos \left (c+\frac {3 d x}{2}\right )-12 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+6 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+66 B \sin \left (\frac {d x}{2}\right )-36 C \sin \left (\frac {d x}{2}\right )-30 B \sin \left (c+\frac {d x}{2}\right )+24 C \sin \left (c+\frac {d x}{2}\right )+41 B \sin \left (c+\frac {3 d x}{2}\right )-20 C \sin \left (c+\frac {3 d x}{2}\right )+9 B \sin \left (2 c+\frac {3 d x}{2}\right )+3 B \sin \left (2 c+\frac {5 d x}{2}\right )+3 B \sin \left (3 c+\frac {5 d x}{2}\right )\right )}{12 a^2 d (1+\cos (c+d x))^2} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (3 B \cos \left (2 d x +2 c \right )+28 B \cos \left (d x +c \right )+23 B +2 C \right )-20 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-24 x d \left (B -\frac {C}{2}\right )}{12 a^{2} d}\) | \(73\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {4 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-4 \left (2 B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(108\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {4 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-4 \left (2 B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(108\) |
risch | \(-\frac {2 B x}{a^{2}}+\frac {x C}{a^{2}}-\frac {i B \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}+\frac {i B \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}+\frac {2 i \left (9 B \,{\mathrm e}^{2 i \left (d x +c \right )}-6 C \,{\mathrm e}^{2 i \left (d x +c \right )}+15 B \,{\mathrm e}^{i \left (d x +c \right )}-9 C \,{\mathrm e}^{i \left (d x +c \right )}+8 B -5 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(130\) |
norman | \(\frac {\frac {\left (2 B -C \right ) x}{a}+\frac {\left (2 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {\left (B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}-\frac {\left (2 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {\left (2 B -C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {3 \left (3 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (7 B -4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {\left (7 B -4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}+\frac {\left (14 B -5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a}\) | \(245\) |
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Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {3 \, {\left (2 \, B - C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, B - C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (2 \, B - C\right )} d x - {\left (3 \, B \cos \left (d x + c\right )^{2} + {\left (14 \, B - 5 \, C\right )} \cos \left (d x + c\right ) + 10 \, B - 4 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (94) = 188\).
Time = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.95 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {6 \, {\left (d x + c\right )} {\left (2 \, B - C\right )}}{a^{2}} - \frac {12 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}} + \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
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Time = 15.80 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B-C}{a^2}+\frac {3\,B-C}{2\,a^2}\right )}{d}-\frac {x\,\left (2\,B-C\right )}{a^2}+\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d} \]
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