Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d} \]
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Time = 0.21 (sec) , antiderivative size = 73, 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, 76} \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \]
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Rule 12
Rule 76
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^7(c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^3 x^2 (-a+x)}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int (-a-x)^3 x^2 (-a+x) \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int \left (a^4 x^2+2 a^3 x^3-2 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = -\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.73 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {4 (17 \cos (c+d x)+10 \cos (2 (c+d x))+3 (4+\cos (3 (c+d x)))) \sin ^8\left (\frac {1}{2} (c+d x)\right )}{21 a^2 d} \]
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Time = 0.85 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(49\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(49\) |
parallelrisch | \(\frac {126 \cos \left (2 d x +2 c \right )-368-14 \cos \left (6 d x +6 c \right )-231 \cos \left (d x +c \right )-49 \cos \left (3 d x +3 c \right )+21 \cos \left (5 d x +5 c \right )+3 \cos \left (7 d x +7 c \right )}{1344 a^{2} d}\) | \(74\) |
risch | \(-\frac {11 \cos \left (d x +c \right )}{64 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{2}}-\frac {\cos \left (6 d x +6 c \right )}{96 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{64 d \,a^{2}}-\frac {7 \cos \left (3 d x +3 c \right )}{192 d \,a^{2}}+\frac {3 \cos \left (2 d x +2 c \right )}{32 d \,a^{2}}\) | \(101\) |
norman | \(\frac {-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d a}-\frac {8}{21 a d}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}-\frac {40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d a}-\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} a}\) | \(124\) |
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, a^{2} d} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (65) = 130\).
Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {8 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {42 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{21 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}+\frac {{\cos \left (c+d\,x\right )}^6}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}}{d} \]
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