Integrand size = 21, antiderivative size = 131 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{d}-\frac {2 a^3 \cos ^4(c+d x)}{d}+\frac {a^3 \cos ^6(c+d x)}{2 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^6(c+d x)}{2 d}-\frac {2 a^3 \cos ^4(c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}+\frac {8 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \sin ^4(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^3 (-a-x)^3 (-a+x)^6}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^3 (-a+x)^6}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-8 a^6-\frac {a^9}{x^3}+\frac {3 a^8}{x^2}+6 a^5 x+6 a^4 x^2-8 a^3 x^3+3 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^4 d} \\ & = \frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{d}-\frac {2 a^3 \cos ^4(c+d x)}{d}+\frac {a^3 \cos ^6(c+d x)}{2 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {a^3 (427+14014 \cos (c+d x)-210 \cos (2 (c+d x))+2548 \cos (3 (c+d x))+196 \cos (4 (c+d x))-188 \cos (5 (c+d x))-56 \cos (6 (c+d x))+9 \cos (7 (c+d x))+7 \cos (8 (c+d x))+\cos (9 (c+d x))) \sec ^2(c+d x)}{1792 d} \]
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Time = 2.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {a^{3} \left (196 \cos \left (4 d x +4 c \right )+2548 \cos \left (3 d x +3 c \right )+14014 \cos \left (d x +c \right )+7 \cos \left (8 d x +8 c \right )+9 \cos \left (7 d x +7 c \right )-56 \cos \left (6 d x +6 c \right )-188 \cos \left (5 d x +5 c \right )+7800 \cos \left (2 d x +2 c \right )+\cos \left (9 d x +9 c \right )+8437\right )}{896 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(118\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(214\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(214\) |
parts | \(-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(222\) |
risch | \(-\frac {29 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {29 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{128 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {5 a^{3} \cos \left (4 d x +4 c \right )}{32 d}\) | \(225\) |
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {32 \, a^{3} \cos \left (d x + c\right )^{9} + 112 \, a^{3} \cos \left (d x + c\right )^{8} - 448 \, a^{3} \cos \left (d x + c\right )^{6} - 448 \, a^{3} \cos \left (d x + c\right )^{5} + 672 \, a^{3} \cos \left (d x + c\right )^{4} + 1792 \, a^{3} \cos \left (d x + c\right )^{3} - 203 \, a^{3} \cos \left (d x + c\right )^{2} + 672 \, a^{3} \cos \left (d x + c\right ) + 112 \, a^{3}}{224 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{7} + 7 \, a^{3} \cos \left (d x + c\right )^{6} - 28 \, a^{3} \cos \left (d x + c\right )^{4} - 28 \, a^{3} \cos \left (d x + c\right )^{3} + 42 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3} \cos \left (d x + c\right ) + \frac {7 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{14 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, {\left (\frac {7 \, {\left (3 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} - \frac {43 \, a^{3} - \frac {273 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {672 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {630 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {343 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {105 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}\right )}}{7 \, d} \]
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Time = 13.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+8\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-2\,a^3\,{\cos \left (c+d\,x\right )}^3-2\,a^3\,{\cos \left (c+d\,x\right )}^4+\frac {a^3\,{\cos \left (c+d\,x\right )}^6}{2}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}}{d} \]
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