Integrand size = 31, antiderivative size = 62 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d}+\frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))} \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2915, 78} \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))}+\frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d} \]
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Rule 78
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^3 \text {Subst}\left (\int \frac {(a+x) \left (A+\frac {B x}{a}\right )}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^3 \text {Subst}\left (\int \left (\frac {B}{a}+\frac {2 a (A+B)}{(a-x)^2}+\frac {-A-3 B}{a-x}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d}+\frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 \left ((A+3 B) \log (1-\sin (c+d x))-\frac {2 (A+B)}{-1+\sin (c+d x)}+B \sin (c+d x)\right )}{d} \]
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Time = 0.46 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.65
method | result | size |
parallelrisch | \(-\frac {\left (\left (\sin \left (d x +c \right )-1\right ) \left (A +3 B \right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\sin \left (d x +c \right )-1\right ) \left (A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {B \cos \left (2 d x +2 c \right )}{2}+\sin \left (d x +c \right ) \left (2 A +3 B \right )-\frac {B}{2}\right ) a^{3}}{d \left (\sin \left (d x +c \right )-1\right )}\) | \(102\) |
risch | \(-i x \,a^{3} A -3 i x \,a^{3} B -\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3}}{2 d}+\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )} B}{2 d}-\frac {2 i a^{3} A c}{d}-\frac {6 i a^{3} B c}{d}-\frac {4 i a^{3} {\mathrm e}^{i \left (d x +c \right )} \left (A +B \right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{2}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}+\frac {6 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}\) | \(157\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \,a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+\frac {3 A \,a^{3}}{2 \cos \left (d x +c \right )^{2}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {B \,a^{3}}{2 \cos \left (d x +c \right )^{2}}}{d}\) | \(273\) |
default | \(\frac {A \,a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \,a^{3} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+\frac {3 A \,a^{3}}{2 \cos \left (d x +c \right )^{2}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {B \,a^{3}}{2 \cos \left (d x +c \right )^{2}}}{d}\) | \(273\) |
norman | \(\frac {\frac {48 A \,a^{3}+48 B \,a^{3}}{4 d}+\frac {\left (48 A \,a^{3}+48 B \,a^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (64 A \,a^{3}+64 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {\left (64 A \,a^{3}+64 B \,a^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {\left (80 A \,a^{3}+80 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (80 A \,a^{3}+80 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {2 a^{3} \left (2 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{3} \left (2 A +3 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{3} \left (10 A +9 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{3} \left (10 A +9 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (10 A +11 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (10 A +11 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2 a^{3} \left (A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a^{3} \left (A +3 B \right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(403\) |
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.44 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {B a^{3} \cos \left (d x + c\right )^{2} + B a^{3} \sin \left (d x + c\right ) + {\left (2 \, A + B\right )} a^{3} - {\left ({\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right ) - {\left (A + 3 \, B\right )} a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \]
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\[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=a^{3} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {{\left (A + 3 \, B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + B a^{3} \sin \left (d x + c\right ) - \frac {2 \, {\left (A + B\right )} a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (63) = 126\).
Time = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.68 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, {\left (A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} + 3 \, B a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 22 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{3} + 9 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}}}{d} \]
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Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (A\,a^3+3\,B\,a^3\right )-\frac {2\,A\,a^3+2\,B\,a^3}{\sin \left (c+d\,x\right )-1}+B\,a^3\,\sin \left (c+d\,x\right )}{d} \]
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