Integrand size = 31, antiderivative size = 115 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^3}{7 d}+\frac {2 (4 A-3 B) \sec ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{35 d}+\frac {3 a^3 (4 A-3 B) \tan (c+d x)}{35 d}+\frac {a^3 (4 A-3 B) \tan ^3(c+d x)}{35 d} \]
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Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2934, 2755, 3852} \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (4 A-3 B) \tan ^3(c+d x)}{35 d}+\frac {3 a^3 (4 A-3 B) \tan (c+d x)}{35 d}+\frac {2 (4 A-3 B) \sec ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^3}{7 d} \]
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Rule 2755
Rule 2934
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^3}{7 d}+\frac {1}{7} (a (4 A-3 B)) \int \sec ^6(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^3}{7 d}+\frac {2 (4 A-3 B) \sec ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{35 d}+\frac {1}{35} \left (3 a^3 (4 A-3 B)\right ) \int \sec ^4(c+d x) \, dx \\ & = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^3}{7 d}+\frac {2 (4 A-3 B) \sec ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{35 d}-\frac {\left (3 a^3 (4 A-3 B)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^3}{7 d}+\frac {2 (4 A-3 B) \sec ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{35 d}+\frac {3 a^3 (4 A-3 B) \tan (c+d x)}{35 d}+\frac {a^3 (4 A-3 B) \tan ^3(c+d x)}{35 d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.17 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (35 B+14 (4 A-3 B) \cos (2 (c+d x))+(-4 A+3 B) \cos (4 (c+d x))+56 A \sin (c+d x)-42 B \sin (c+d x)-24 A \sin (3 (c+d x))+18 B \sin (3 (c+d x)))}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {4 i a^{3} \left (56 i A \,{\mathrm e}^{3 i \left (d x +c \right )}-42 i B \,{\mathrm e}^{3 i \left (d x +c \right )}+35 B \,{\mathrm e}^{4 i \left (d x +c \right )}-24 i A \,{\mathrm e}^{i \left (d x +c \right )}+56 A \,{\mathrm e}^{2 i \left (d x +c \right )}+18 i B \,{\mathrm e}^{i \left (d x +c \right )}-42 B \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A +3 B \right )}{35 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{7} d}\) | \(133\) |
parallelrisch | \(-\frac {2 a^{3} \left (A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 A -2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (-A +B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (-A -8 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (11 A +3 B \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (-43 A +6 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35}+\frac {13 A}{35}-\frac {B}{35}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(167\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+\frac {3 A \,a^{3}}{7 \cos \left (d x +c \right )^{7}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(435\) |
default | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+\frac {3 A \,a^{3}}{7 \cos \left (d x +c \right )^{7}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(435\) |
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Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.27 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {2 \, {\left (4 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 9 \, {\left (4 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (3 \, A - 4 \, B\right )} a^{3} + {\left (6 \, {\left (4 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - 5 \, {\left (4 \, A - 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{35 \, {\left (3 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (107) = 214\).
Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.98 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} A a^{3} + {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{3} + {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} B a^{3} + {\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} B a^{3} - \frac {{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} A a^{3}}{\cos \left (d x + c\right )^{7}} - \frac {3 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} B a^{3}}{\cos \left (d x + c\right )^{7}} + \frac {15 \, A a^{3}}{\cos \left (d x + c\right )^{7}} + \frac {5 \, B a^{3}}{\cos \left (d x + c\right )^{7}}}{35 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (107) = 214\).
Time = 0.36 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.26 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {\frac {35 \, {\left (A a^{3} - B a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} + \frac {525 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 35 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1960 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4025 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 665 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 4480 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1120 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3143 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 791 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1176 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 392 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 243 \, A a^{3} - 51 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{7}}}{280 \, d} \]
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Time = 11.60 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.85 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {35\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {91\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}+A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-\frac {35\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {21\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-\frac {3\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{4}-\frac {233\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {121\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {61\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {13\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {61\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {23\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {37\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}\right )}{280\,d\,\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^7} \]
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