Integrand size = 23, antiderivative size = 114 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \sin (c+d x)}{d}-\frac {2 a (2 a-b) \sin ^3(c+d x)}{3 d}+\frac {\left (6 a^2-6 a b+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac {2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}+\frac {(a-b)^2 \sin ^9(c+d x)}{9 d} \]
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Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3757, 380} \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\left (6 a^2-6 a b+b^2\right ) \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin (c+d x)}{d}+\frac {(a-b)^2 \sin ^9(c+d x)}{9 d}-\frac {2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}-\frac {2 a (2 a-b) \sin ^3(c+d x)}{3 d} \]
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Rule 380
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1-x^2\right )^2 \left (a-(a-b) x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (a^2-2 a (2 a-b) x^2+\left (6 a^2-6 a b+b^2\right ) x^4-2 \left (2 a^2-3 a b+b^2\right ) x^6+(a-b)^2 x^8\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {a^2 \sin (c+d x)}{d}-\frac {2 a (2 a-b) \sin ^3(c+d x)}{3 d}+\frac {\left (6 a^2-6 a b+b^2\right ) \sin ^5(c+d x)}{5 d}-\frac {2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}+\frac {(a-b)^2 \sin ^9(c+d x)}{9 d} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {630 \left (63 a^2+14 a b+3 b^2\right ) \sin (c+d x)+420 \left (21 a^2-b^2\right ) \sin (3 (c+d x))+252 \left (9 a^2-4 a b-b^2\right ) \sin (5 (c+d x))+45 (a-b) (9 a-b) \sin (7 (c+d x))+35 (a-b)^2 \sin (9 (c+d x))}{80640 d} \]
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Time = 121.92 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.61
method | result | size |
derivativedivides | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+2 a b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+\frac {a^{2} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(183\) |
default | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+2 a b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+\frac {a^{2} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(183\) |
risch | \(\frac {63 a^{2} \sin \left (d x +c \right )}{128 d}+\frac {7 \sin \left (d x +c \right ) a b}{64 d}+\frac {3 \sin \left (d x +c \right ) b^{2}}{128 d}+\frac {\sin \left (9 d x +9 c \right ) a^{2}}{2304 d}-\frac {\sin \left (9 d x +9 c \right ) a b}{1152 d}+\frac {\sin \left (9 d x +9 c \right ) b^{2}}{2304 d}+\frac {9 \sin \left (7 d x +7 c \right ) a^{2}}{1792 d}-\frac {5 \sin \left (7 d x +7 c \right ) a b}{896 d}+\frac {\sin \left (7 d x +7 c \right ) b^{2}}{1792 d}+\frac {9 \sin \left (5 d x +5 c \right ) a^{2}}{320 d}-\frac {\sin \left (5 d x +5 c \right ) a b}{80 d}-\frac {\sin \left (5 d x +5 c \right ) b^{2}}{320 d}+\frac {7 \sin \left (3 d x +3 c \right ) a^{2}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) b^{2}}{192 d}\) | \(227\) |
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {{\left (35 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{2} + a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (16 \, a^{2} + 4 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (16 \, a^{2} + 4 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 128 \, a^{2} + 32 \, a b + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]
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Timed out. \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {35 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{9} - 90 \, {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{7} + 63 \, {\left (6 \, a^{2} - 6 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{5} - 210 \, {\left (2 \, a^{2} - a b\right )} \sin \left (d x + c\right )^{3} + 315 \, a^{2} \sin \left (d x + c\right )}{315 \, d} \]
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Timed out. \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\text {Timed out} \]
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Time = 12.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.65 \[ \int \cos ^9(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\frac {63\,a^2\,\sin \left (c+d\,x\right )}{128}+\frac {3\,b^2\,\sin \left (c+d\,x\right )}{128}+\frac {7\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{64}+\frac {9\,a^2\,\sin \left (5\,c+5\,d\,x\right )}{320}+\frac {9\,a^2\,\sin \left (7\,c+7\,d\,x\right )}{1792}+\frac {a^2\,\sin \left (9\,c+9\,d\,x\right )}{2304}-\frac {b^2\,\sin \left (3\,c+3\,d\,x\right )}{192}-\frac {b^2\,\sin \left (5\,c+5\,d\,x\right )}{320}+\frac {b^2\,\sin \left (7\,c+7\,d\,x\right )}{1792}+\frac {b^2\,\sin \left (9\,c+9\,d\,x\right )}{2304}+\frac {7\,a\,b\,\sin \left (c+d\,x\right )}{64}-\frac {a\,b\,\sin \left (5\,c+5\,d\,x\right )}{80}-\frac {5\,a\,b\,\sin \left (7\,c+7\,d\,x\right )}{896}-\frac {a\,b\,\sin \left (9\,c+9\,d\,x\right )}{1152}}{d} \]
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