Integrand size = 29, antiderivative size = 81 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {B (a+a \sin (c+d x))^3}{3 d} \]
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Time = 0.07 (sec) , antiderivative size = 81, 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 = {2915, 78} \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {B (a \sin (c+d x)+a)^3}{3 d} \]
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Rule 78
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a \text {Subst}\left (\int \frac {(a+x)^2 \left (A+\frac {B x}{a}\right )}{a-x} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (-3 a (A+B)+\frac {4 a^2 (A+B)}{a-x}-(A+B) x-\frac {B (a+x)^2}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {B (a+a \sin (c+d x))^3}{3 d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \left (24 (A+B) \log (1-\sin (c+d x))+6 (3 A+4 B) \sin (c+d x)+3 (A+3 B) \sin ^2(c+d x)+2 B \sin ^3(c+d x)\right )}{6 d} \]
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Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {4 \left (\left (A +B \right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-A -B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {\left (A +3 B \right ) \cos \left (2 d x +2 c \right )}{16}+\frac {B \sin \left (3 d x +3 c \right )}{48}+\frac {\left (-3 A -\frac {17 B}{4}\right ) \sin \left (d x +c \right )}{4}-\frac {A}{16}-\frac {3 B}{16}\right ) a^{3}}{d}\) | \(96\) |
derivativedivides | \(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 A \,a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-3 A \,a^{3} \ln \left (\cos \left (d x +c \right )\right )+3 B \,a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-B \,a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(198\) |
default | \(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 A \,a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-3 A \,a^{3} \ln \left (\cos \left (d x +c \right )\right )+3 B \,a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-B \,a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(198\) |
risch | \(4 i x \,a^{3} A +4 i x \,a^{3} B +\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{3}}{2 d}+\frac {17 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3}}{8 d}-\frac {3 i a^{3} {\mathrm e}^{-i \left (d x +c \right )} A}{2 d}-\frac {17 i a^{3} {\mathrm e}^{-i \left (d x +c \right )} B}{8 d}+\frac {8 i a^{3} A c}{d}+\frac {8 i a^{3} B c}{d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{12 d}+\frac {a^{3} \cos \left (2 d x +2 c \right ) A}{4 d}+\frac {3 a^{3} \cos \left (2 d x +2 c \right ) B}{4 d}\) | \(214\) |
norman | \(\frac {-\frac {2 \left (A \,a^{3}+3 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (A \,a^{3}+3 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (2 A \,a^{3}+6 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{3} \left (3 A +4 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{3} \left (3 A +4 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{3} \left (27 A +40 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{3} \left (27 A +40 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {8 a^{3} \left (A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {4 a^{3} \left (A +B \right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(254\) |
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {3 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - 24 \, {\left (A + B\right )} a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{3} \cos \left (d x + c\right )^{2} - {\left (9 \, A + 13 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \sin ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {2 \, B a^{3} \sin \left (d x + c\right )^{3} + 3 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{2} + 24 \, {\left (A + B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, {\left (3 \, A + 4 \, B\right )} a^{3} \sin \left (d x + c\right )}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (77) = 154\).
Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.57 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {2 \, {\left (6 \, {\left (A a^{3} + B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 12 \, {\left (A a^{3} + B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {11 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 42 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 28 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, A a^{3} + 11 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}\right )}}{3 \, d} \]
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Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3\,\left (A+2\,B\right )}{2}+\frac {B\,a^3}{2}\right )+\sin \left (c+d\,x\right )\,\left (a^3\,\left (A+2\,B\right )+a^3\,\left (2\,A+B\right )+B\,a^3\right )+\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (4\,A\,a^3+4\,B\,a^3\right )+\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
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