Integrand size = 21, antiderivative size = 92 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {(A+C) \sin (c+d x)}{d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {C \sin ^9(c+d x)}{9 d} \]
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Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3092, 380} \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {C \sin ^9(c+d x)}{9 d} \]
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Rule 380
Rule 3092
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-x^2\right )^3 \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(3 A+4 C) x^2+3 (A+2 C) x^4-(A+4 C) x^6+C x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {(A+C) \sin (c+d x)}{d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {C \sin ^9(c+d x)}{9 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.45 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {A \sin (c+d x)}{d}+\frac {C \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{d}-\frac {4 C \sin ^3(c+d x)}{3 d}+\frac {3 A \sin ^5(c+d x)}{5 d}+\frac {6 C \sin ^5(c+d x)}{5 d}-\frac {A \sin ^7(c+d x)}{7 d}-\frac {4 C \sin ^7(c+d x)}{7 d}+\frac {C \sin ^9(c+d x)}{9 d} \]
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Time = 4.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\frac {8820 \left (A +C \right ) \sin \left (3 d x +3 c \right )+252 \left (7 A +9 C \right ) \sin \left (5 d x +5 c \right )+45 \left (4 A +9 C \right ) \sin \left (7 d x +7 c \right )+35 C \sin \left (9 d x +9 c \right )+44100 \left (A +\frac {9 C}{10}\right ) \sin \left (d x +c \right )}{80640 d}\) | \(82\) |
derivativedivides | \(\frac {\frac {C \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}+\frac {A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(94\) |
default | \(\frac {\frac {C \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}+\frac {A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(94\) |
parts | \(\frac {A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {C \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9 d}\) | \(96\) |
risch | \(\frac {35 \sin \left (d x +c \right ) A}{64 d}+\frac {63 C \sin \left (d x +c \right )}{128 d}+\frac {C \sin \left (9 d x +9 c \right )}{2304 d}+\frac {\sin \left (7 d x +7 c \right ) A}{448 d}+\frac {9 \sin \left (7 d x +7 c \right ) C}{1792 d}+\frac {7 \sin \left (5 d x +5 c \right ) A}{320 d}+\frac {9 \sin \left (5 d x +5 c \right ) C}{320 d}+\frac {7 \sin \left (3 d x +3 c \right ) A}{64 d}+\frac {7 \sin \left (3 d x +3 c \right ) C}{64 d}\) | \(131\) |
norman | \(\frac {\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 \left (3 A +2 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (3 A +2 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (17 A +19 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (17 A +19 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (221 A +142 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {8 \left (221 A +142 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 \left (4617 A +5329 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(215\) |
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Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (35 \, C \cos \left (d x + c\right )^{8} + 5 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{2} + 144 \, A + 128 \, C\right )} \sin \left (d x + c\right )}{315 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (80) = 160\).
Time = 0.92 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.16 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {16 A \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {128 C \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 C \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 C \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 C \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {35 \, C \sin \left (d x + c\right )^{9} - 45 \, {\left (A + 4 \, C\right )} \sin \left (d x + c\right )^{7} + 189 \, {\left (A + 2 \, C\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (3 \, A + 4 \, C\right )} \sin \left (d x + c\right )^{3} + 315 \, {\left (A + C\right )} \sin \left (d x + c\right )}{315 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (4 \, A + 9 \, C\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A + 9 \, C\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (A + C\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {7 \, {\left (10 \, A + 9 \, C\right )} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 0.82 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80 \[ \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\frac {C\,{\sin \left (c+d\,x\right )}^9}{9}+\left (-\frac {A}{7}-\frac {4\,C}{7}\right )\,{\sin \left (c+d\,x\right )}^7+\left (\frac {3\,A}{5}+\frac {6\,C}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-A-\frac {4\,C}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (A+C\right )\,\sin \left (c+d\,x\right )}{d} \]
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