Integrand size = 31, antiderivative size = 107 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))^3}{5 d}+\frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d \left (a^2-a^2 \sin (c+d x)\right )} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2934, 2749, 2729, 2727} \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^3}{5 d} \]
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Rule 2727
Rule 2729
Rule 2749
Rule 2934
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))^3}{5 d}+\frac {1}{5} (a (2 A-3 B)) \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = \frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))^3}{5 d}+\frac {1}{5} \left (a^5 (2 A-3 B)\right ) \int \frac {1}{(a-a \sin (c+d x))^2} \, dx \\ & = \frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))^3}{5 d}+\frac {1}{15} \left (a^4 (2 A-3 B)\right ) \int \frac {1}{a-a \sin (c+d x)} \, dx \\ & = \frac {a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {a^4 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))}+\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))^3}{5 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (-16 A+9 B+(2 A-3 B) \cos (2 (c+d x))+6 (2 A-3 B) \sin (c+d x))}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
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Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(-\frac {2 a^{3} \left (A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B -2 A \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {8 A}{3}-B \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {4 A}{3}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 A}{15}-\frac {B}{5}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}\) | \(93\) |
risch | \(\frac {2 i a^{3} \left (20 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-15 i B \,{\mathrm e}^{2 i \left (d x +c \right )}+15 B \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i A +10 A \,{\mathrm e}^{i \left (d x +c \right )}+3 i B -15 B \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{5}}\) | \(95\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )^{5}}+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+\frac {3 A \,a^{3}}{5 \cos \left (d x +c \right )^{5}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(333\) |
default | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )^{5}}+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+\frac {3 A \,a^{3}}{5 \cos \left (d x +c \right )^{5}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(333\) |
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Time = 0.25 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.76 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {{\left (2 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} - 2 \, {\left (2 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - 3 \, {\left (3 \, A - 2 \, B\right )} a^{3} \cos \left (d x + c\right ) - 3 \, {\left (A + B\right )} a^{3} + {\left ({\left (2 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right ) - 3 \, {\left (A + B\right )} a^{3}\right )} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.76 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {3 \, B a^{3} \tan \left (d x + c\right )^{5} + {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 3 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} A a^{3} + 3 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} B a^{3} - \frac {{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} A a^{3}}{\cos \left (d x + c\right )^{5}} - \frac {3 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} B a^{3}}{\cos \left (d x + c\right )^{5}} + \frac {9 \, A a^{3}}{\cos \left (d x + c\right )^{5}} + \frac {3 \, B a^{3}}{\cos \left (d x + c\right )^{5}}}{15 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.36 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 20 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, A a^{3} - 3 \, B a^{3}\right )}}{15 \, d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{5}} \]
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Time = 11.83 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {\sqrt {2}\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,B-\frac {53\,A}{4}+4\,A\,\cos \left (c+d\,x\right )+\frac {3\,B\,\cos \left (c+d\,x\right )}{2}+\frac {25\,A\,\sin \left (c+d\,x\right )}{2}-\frac {15\,B\,\sin \left (c+d\,x\right )}{2}+\frac {9\,A\,\cos \left (2\,c+2\,d\,x\right )}{4}-\frac {3\,B\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {5\,A\,\sin \left (2\,c+2\,d\,x\right )}{4}\right )}{60\,d\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^5} \]
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