Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d} \]
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Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2914, 2644, 30, 2645, 276} \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^6(c+d x)}{6 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rule 30
Rule 276
Rule 2644
Rule 2645
Rule 2914
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^7(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin ^5(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int x^5 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\sin ^6(c+d x)}{6 a d}+\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^3(c+d x)}{3 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {4 (123+197 \cos (c+d x)+85 \cos (2 (c+d x))+15 \cos (3 (c+d x))) \sin ^8\left (\frac {1}{2} (c+d x)\right )}{105 a d} \]
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Time = 0.66 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{6}-\frac {2 \cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(69\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{6}-\frac {2 \cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(69\) |
parallelrisch | \(\frac {-525 \cos \left (2 d x +2 c \right )+862+210 \cos \left (4 d x +4 c \right )+525 \cos \left (d x +c \right )+35 \cos \left (3 d x +3 c \right )-63 \cos \left (5 d x +5 c \right )-35 \cos \left (6 d x +6 c \right )+15 \cos \left (7 d x +7 c \right )}{6720 d a}\) | \(85\) |
norman | \(\frac {\frac {16}{105 a d}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d a}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d a}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d a}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(102\) |
risch | \(\frac {5 \cos \left (d x +c \right )}{64 a d}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}-\frac {\cos \left (6 d x +6 c \right )}{192 a d}-\frac {3 \cos \left (5 d x +5 c \right )}{320 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {\cos \left (3 d x +3 c \right )}{192 a d}-\frac {5 \cos \left (2 d x +2 c \right )}{64 a d}\) | \(118\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, a d} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, a d} \]
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Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {16 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )}}{105 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15 \[ \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a}+\frac {2\,{\cos \left (c+d\,x\right )}^5}{5\,a}+\frac {{\cos \left (c+d\,x\right )}^6}{6\,a}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}}{d} \]
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