Integrand size = 33, antiderivative size = 156 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {3 (5 A+4 C) x}{8 a}-\frac {(4 A+3 C) \sin (c+d x)}{a d}+\frac {3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(4 A+3 C) \sin ^3(c+d x)}{3 a d} \]
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Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4170, 3872, 2715, 8, 2713} \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {(4 A+3 C) \sin ^3(c+d x)}{3 a d}-\frac {(4 A+3 C) \sin (c+d x)}{a d}+\frac {(5 A+4 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 (5 A+4 C) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac {3 x (5 A+4 C)}{8 a} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {\int \cos ^4(c+d x) (-a (5 A+4 C)+a (4 A+3 C) \sec (c+d x)) \, dx}{a^2} \\ & = -\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {(4 A+3 C) \int \cos ^3(c+d x) \, dx}{a}+\frac {(5 A+4 C) \int \cos ^4(c+d x) \, dx}{a} \\ & = \frac {(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(3 (5 A+4 C)) \int \cos ^2(c+d x) \, dx}{4 a}+\frac {(4 A+3 C) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d} \\ & = -\frac {(4 A+3 C) \sin (c+d x)}{a d}+\frac {3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(4 A+3 C) \sin ^3(c+d x)}{3 a d}+\frac {(3 (5 A+4 C)) \int 1 \, dx}{8 a} \\ & = \frac {3 (5 A+4 C) x}{8 a}-\frac {(4 A+3 C) \sin (c+d x)}{a d}+\frac {3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(4 A+3 C) \sin ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 2.58 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (72 (5 A+4 C) d x \cos \left (\frac {d x}{2}\right )+72 (5 A+4 C) d x \cos \left (c+\frac {d x}{2}\right )-552 A \sin \left (\frac {d x}{2}\right )-480 C \sin \left (\frac {d x}{2}\right )-168 A \sin \left (c+\frac {d x}{2}\right )-96 C \sin \left (c+\frac {d x}{2}\right )-120 A \sin \left (c+\frac {3 d x}{2}\right )-72 C \sin \left (c+\frac {3 d x}{2}\right )-120 A \sin \left (2 c+\frac {3 d x}{2}\right )-72 C \sin \left (2 c+\frac {3 d x}{2}\right )+40 A \sin \left (2 c+\frac {5 d x}{2}\right )+24 C \sin \left (2 c+\frac {5 d x}{2}\right )+40 A \sin \left (3 c+\frac {5 d x}{2}\right )+24 C \sin \left (3 c+\frac {5 d x}{2}\right )-5 A \sin \left (3 c+\frac {7 d x}{2}\right )-5 A \sin \left (4 c+\frac {7 d x}{2}\right )+3 A \sin \left (4 c+\frac {9 d x}{2}\right )+3 A \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{192 a d (1+\cos (c+d x))} \]
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Time = 0.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {\left (\left (38 A +24 C \right ) \cos \left (2 d x +2 c \right )-2 A \cos \left (3 d x +3 c \right )+3 A \cos \left (4 d x +4 c \right )+\left (-82 A -48 C \right ) \cos \left (d x +c \right )-221 A -168 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+180 \left (A +\frac {4 C}{5}\right ) x d}{96 d a}\) | \(91\) |
derivativedivides | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {25 A}{8}-\frac {3 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (-\frac {115 A}{24}-\frac {7 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {109 A}{24}-\frac {5 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-\frac {7 A}{8}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (5 A +4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(144\) |
default | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {25 A}{8}-\frac {3 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+2 \left (-\frac {115 A}{24}-\frac {7 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-\frac {109 A}{24}-\frac {5 C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-\frac {7 A}{8}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (5 A +4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(144\) |
risch | \(\frac {15 A x}{8 a}+\frac {3 x C}{2 a}+\frac {7 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a d}-\frac {7 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {A \sin \left (4 d x +4 c \right )}{32 a d}-\frac {A \sin \left (3 d x +3 c \right )}{12 a d}+\frac {A \sin \left (2 d x +2 c \right )}{2 a d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 a d}\) | \(210\) |
norman | \(\frac {-\frac {3 \left (5 A +4 C \right ) x}{8 a}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{a d}-\frac {9 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}+\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}+\frac {9 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}+\frac {3 \left (5 A +4 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}+\frac {\left (5 A +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a d}-\frac {2 \left (8 A +9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}+\frac {\left (11 A +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (31 A +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {\left (37 A +24 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(316\) |
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Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {9 \, {\left (5 \, A + 4 \, C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (5 \, A + 4 \, C\right )} d x + {\left (6 \, A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} + {\left (13 \, A + 12 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (19 \, A + 12 \, C\right )} \cos \left (d x + c\right ) - 64 \, A - 48 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
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\[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {A \cos ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (148) = 296\).
Time = 0.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.25 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=-\frac {A {\left (\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, C {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\frac {9 \, {\left (d x + c\right )} {\left (5 \, A + 4 \, C\right )}}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (75 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 115 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
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Time = 14.94 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {15\,A\,x}{8\,a}+\frac {3\,C\,x}{2\,a}-\frac {7\,A\,\sin \left (c+d\,x\right )}{4\,a\,d}-\frac {C\,\sin \left (c+d\,x\right )}{a\,d}+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{2\,a\,d}-\frac {A\,\sin \left (3\,c+3\,d\,x\right )}{12\,a\,d}+\frac {A\,\sin \left (4\,c+4\,d\,x\right )}{32\,a\,d}-\frac {A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,a\,d}-\frac {C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]
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