Integrand size = 21, antiderivative size = 98 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {2 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3957, 2786, 76} \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^2(c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {2 a^3 \log (\cos (c+d x))}{d} \]
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Rule 76
Rule 2786
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \tan ^3(c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x) (-a+x)^4}{x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a^2-\frac {a^5}{x^3}+\frac {3 a^4}{x^2}-\frac {2 a^3}{x}+3 a x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {2 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {a^3 (-41+226 \cos (c+d x)+29 \cos (3 (c+d x))+9 \cos (4 (c+d x))+\cos (5 (c+d x))-48 \log (\cos (c+d x))-8 \cos (2 (c+d x)) (7+6 \log (\cos (c+d x)))) \sec ^2(c+d x)}{48 d} \]
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Time = 2.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(114\) |
default | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(114\) |
parts | \(-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(122\) |
parallelrisch | \(\frac {a^{3} \left (48 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+226 \cos \left (d x +c \right )+116 \cos \left (2 d x +2 c \right )+29 \cos \left (3 d x +3 c \right )+9 \cos \left (4 d x +4 c \right )+\cos \left (5 d x +5 c \right )+131\right )}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(173\) |
norman | \(\frac {\frac {32 a^{3}}{3 d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d}+\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{3} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(180\) |
risch | \(2 i a^{3} x +\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {9 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {9 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {4 i a^{3} c}{d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(194\) |
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Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{3} \cos \left (d x + c\right )^{4} + 24 \, a^{3} \cos \left (d x + c\right )^{3} - 24 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 9 \, a^{3} \cos \left (d x + c\right )^{2} + 36 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=a^{3} \left (\int 3 \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )^{2} + 12 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {3 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{6 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=-\frac {2 \, a^{3} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} + \frac {2 \, a^{3} d^{8} \cos \left (d x + c\right )^{3} + 9 \, a^{3} d^{8} \cos \left (d x + c\right )^{2} + 12 \, a^{3} d^{8} \cos \left (d x + c\right )}{6 \, d^{9}} \]
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Time = 13.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+2\,a^3\,\cos \left (c+d\,x\right )+\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {a^3\,{\cos \left (c+d\,x\right )}^3}{3}-2\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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