Integrand size = 33, antiderivative size = 137 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {(7 A+2 C) x}{2 a^2}-\frac {4 (4 A+C) \sin (c+d x)}{3 a^2 d}+\frac {(7 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {2 (4 A+C) \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
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Time = 0.36 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4170, 4105, 3872, 2715, 8, 2717} \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 (4 A+C) \sin (c+d x)}{3 a^2 d}+\frac {(7 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac {2 (4 A+C) \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {x (7 A+2 C)}{2 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4105
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) (-a (5 A+2 C)+3 a A \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {2 (4 A+C) \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\int \cos ^2(c+d x) \left (-3 a^2 (7 A+2 C)+4 a^2 (4 A+C) \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {2 (4 A+C) \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(4 (4 A+C)) \int \cos (c+d x) \, dx}{3 a^2}+\frac {(7 A+2 C) \int \cos ^2(c+d x) \, dx}{a^2} \\ & = -\frac {4 (4 A+C) \sin (c+d x)}{3 a^2 d}+\frac {(7 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {2 (4 A+C) \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(7 A+2 C) \int 1 \, dx}{2 a^2} \\ & = \frac {(7 A+2 C) x}{2 a^2}-\frac {4 (4 A+C) \sin (c+d x)}{3 a^2 d}+\frac {(7 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {2 (4 A+C) \cos (c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \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. \(281\) vs. \(2(137)=274\).
Time = 2.20 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.05 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (36 (7 A+2 C) d x \cos \left (\frac {d x}{2}\right )+36 (7 A+2 C) d x \cos \left (c+\frac {d x}{2}\right )+84 A d x \cos \left (c+\frac {3 d x}{2}\right )+24 C d x \cos \left (c+\frac {3 d x}{2}\right )+84 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+24 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-381 A \sin \left (\frac {d x}{2}\right )-144 C \sin \left (\frac {d x}{2}\right )+147 A \sin \left (c+\frac {d x}{2}\right )+96 C \sin \left (c+\frac {d x}{2}\right )-239 A \sin \left (c+\frac {3 d x}{2}\right )-80 C \sin \left (c+\frac {3 d x}{2}\right )-63 A \sin \left (2 c+\frac {3 d x}{2}\right )-15 A \sin \left (2 c+\frac {5 d x}{2}\right )-15 A \sin \left (3 c+\frac {5 d x}{2}\right )+3 A \sin \left (3 c+\frac {7 d x}{2}\right )+3 A \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a^2 d (1+\sec (c+d x))^2} \]
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Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {\left (-80 C +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (3 A \cos \left (3 d x +3 c \right )-12 A \cos \left (2 d x +2 c \right )-163 A \cos \left (d x +c \right )-140 A +8 C \right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+168 \left (A +\frac {2 C}{7}\right ) x d}{48 a^{2} d}\) | \(85\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+2 \left (7 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(125\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+2 \left (7 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(125\) |
risch | \(\frac {7 A x}{2 a^{2}}+\frac {x C}{a^{2}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{2} d}+\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{2} d}-\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{2} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{2} d}-\frac {2 i \left (12 A \,{\mathrm e}^{2 i \left (d x +c \right )}+6 C \,{\mathrm e}^{2 i \left (d x +c \right )}+21 A \,{\mathrm e}^{i \left (d x +c \right )}+9 C \,{\mathrm e}^{i \left (d x +c \right )}+11 A +5 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(168\) |
norman | \(\frac {-\frac {\left (7 A +2 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}+\frac {4 \left (4 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {2 \left (5 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}-\frac {\left (7 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {\left (7 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}+\frac {\left (7 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}+\frac {\left (13 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (26 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a}\) | \(243\) |
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Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {3 \, {\left (7 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (7 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (7 \, A + 2 \, C\right )} d x + {\left (3 \, A \cos \left (d x + c\right )^{3} - 6 \, A \cos \left (d x + c\right )^{2} - {\left (43 \, A + 10 \, C\right )} \cos \left (d x + c\right ) - 32 \, A - 8 \, C\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {A {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} + C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (d x + c\right )} {\left (7 \, A + 2 \, C\right )}}{a^{2}} - \frac {6 \, {\left (5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
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Time = 15.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {x\,\left (7\,A+2\,C\right )}{2\,a^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A+C\right )}{2\,a^2}+\frac {2\,A}{a^2}\right )}{d}-\frac {5\,A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \]
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