Integrand size = 29, antiderivative size = 73 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {a (4 A-B) \tan ^3(c+d x)}{15 d} \]
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Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2934, 3852} \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a (4 A-B) \tan ^3(c+d x)}{15 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)}{5 d} \]
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Rule 2934
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}+\frac {1}{5} (a (4 A-B)) \int \sec ^4(c+d x) \, dx \\ & = \frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}-\frac {(a (4 A-B)) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {a (4 A-B) \tan ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (3 (A+B) \sec ^5(c+d x)+15 A \sec ^4(c+d x) \tan (c+d x)-5 (4 A-B) \sec ^2(c+d x) \tan ^3(c+d x)+2 (4 A-B) \tan ^5(c+d x)\right )}{15 d} \]
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Time = 0.50 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(\frac {\frac {a A}{5 \cos \left (d x +c \right )^{5}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {B a}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(102\) |
default | \(\frac {\frac {a A}{5 \cos \left (d x +c \right )^{5}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {B a}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(102\) |
risch | \(-\frac {4 i a \left (24 i A \,{\mathrm e}^{3 i \left (d x +c \right )}-6 i B \,{\mathrm e}^{3 i \left (d x +c \right )}+15 B \,{\mathrm e}^{4 i \left (d x +c \right )}+8 i A \,{\mathrm e}^{i \left (d x +c \right )}+8 A \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,{\mathrm e}^{i \left (d x +c \right )}-2 B \,{\mathrm e}^{2 i \left (d x +c \right )}+4 A -B \right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{5} d}\) | \(131\) |
parallelrisch | \(-\frac {2 \left (A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (-A -2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\left (5 A +B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\left (13 A +8 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {\left (-7 A +3 B \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+\frac {A}{5}+\frac {B}{5}\right ) a}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}\) | \(166\) |
norman | \(\frac {-\frac {2 a A +2 B a}{5 d}-\frac {\left (2 a A +2 B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (4 a A +4 B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (4 a A +4 B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (6 a A +6 B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 a A +8 B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (22 a A +22 B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a A \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (A +2 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a \left (A +2 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a \left (11 A +16 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 a \left (11 A +16 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a \left (19 A +14 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}\) | \(347\) |
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Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {2 \, {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{2} - {\left (A - 4 \, B\right )} a + {\left (2 \, {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.18 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a + {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {3 \, A a}{\cos \left (d x + c\right )^{5}} + \frac {3 \, B a}{\cos \left (d x + c\right )^{5}}}{15 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (67) = 134\).
Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.08 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {5 \, {\left (15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13 \, A a - 7 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} + \frac {165 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 650 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 400 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 20 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 113 \, A a + 13 \, B a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{5}}}{120 \, d} \]
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Time = 11.74 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.07 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {9\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-\frac {15\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+\frac {B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{4}-\frac {73\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {25\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {19\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {3\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {7\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {5\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {3\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}\right )}{120\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^5} \]
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