Integrand size = 31, antiderivative size = 129 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {a^2 (5 A-2 B) \tan (c+d x)}{7 d}+\frac {2 a^2 (5 A-2 B) \tan ^3(c+d x)}{21 d}+\frac {a^2 (5 A-2 B) \tan ^5(c+d x)}{35 d} \]
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Time = 0.10 (sec) , antiderivative size = 129, 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, 2748, 3852} \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (5 A-2 B) \tan ^5(c+d x)}{35 d}+\frac {2 a^2 (5 A-2 B) \tan ^3(c+d x)}{21 d}+\frac {a^2 (5 A-2 B) \tan (c+d x)}{7 d}+\frac {a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
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Rule 2748
Rule 2934
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {1}{7} (a (5 A-2 B)) \int \sec ^6(c+d x) (a+a \sin (c+d x)) \, dx \\ & = \frac {a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {1}{7} \left (a^2 (5 A-2 B)\right ) \int \sec ^6(c+d x) \, dx \\ & = \frac {a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\left (a^2 (5 A-2 B)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d} \\ & = \frac {a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {a^2 (5 A-2 B) \tan (c+d x)}{7 d}+\frac {2 a^2 (5 A-2 B) \tan ^3(c+d x)}{21 d}+\frac {a^2 (5 A-2 B) \tan ^5(c+d x)}{35 d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 \left ((30 A+9 B) \sec ^7(c+d x)+105 A \sec ^6(c+d x) \tan (c+d x)+21 B \sec ^5(c+d x) \tan ^2(c+d x)-35 (5 A-2 B) \sec ^4(c+d x) \tan ^3(c+d x)+28 (5 A-2 B) \sec ^2(c+d x) \tan ^5(c+d x)+8 (-5 A+2 B) \tan ^7(c+d x)\right )}{105 d} \]
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Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {16 \left (-16 B \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+42 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+40 A \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+20 A \,a^{2} {\mathrm e}^{i \left (d x +c \right )}-8 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}+2 i B \,a^{2}-6 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+70 i A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-28 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+15 i A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 i A \,a^{2}\right )}{105 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{7} d}\) | \(188\) |
parallelrisch | \(-\frac {2 \left (A \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B -2 A \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 \left (2 A +B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (-A +\frac {2 B}{5}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 \left (-2 A +\frac {B}{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 \left (19 A -\frac {17 B}{5}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}+\frac {4 \left (-2 A +\frac {11 B}{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (-A -\frac {12 B}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{7}+\frac {2 A}{7}+\frac {3 B}{35}\right ) a^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(206\) |
derivativedivides | \(\frac {A \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+\frac {2 A \,a^{2}}{7 \cos \left (d x +c \right )^{7}}+2 B \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(295\) |
default | \(\frac {A \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+\frac {2 A \,a^{2}}{7 \cos \left (d x +c \right )^{7}}+2 B \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(295\) |
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Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.22 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {16 \, {\left (5 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 8 \, {\left (5 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 5 \, {\left (2 \, A - 5 \, B\right )} a^{2} - {\left (8 \, {\left (5 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 12 \, {\left (5 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 5 \, {\left (5 \, A - 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} A a^{2} + 3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{2} + 2 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} B a^{2} - \frac {3 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} B a^{2}}{\cos \left (d x + c\right )^{7}} + \frac {30 \, A a^{2}}{\cos \left (d x + c\right )^{7}} + \frac {15 \, B a^{2}}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (119) = 238\).
Time = 0.49 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.52 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {\frac {35 \, {\left (9 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, A a^{2} - 5 \, B a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} + \frac {1365 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 210 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5775 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12250 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 175 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 14350 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 910 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10185 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 756 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3955 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 427 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 760 \, A a^{2} - 31 \, B a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{7}}}{840 \, d} \]
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Time = 12.92 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.12 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {25\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {105\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {95\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {15\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}-21\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {105\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {41\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {55\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {9\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {125\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {55\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-\frac {25\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{2}+5\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+\frac {5\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}+\frac {37\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {19\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}+\frac {13\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{4}-B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\right )}{1680\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^7} \]
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