Integrand size = 29, antiderivative size = 129 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {11 x}{16 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d} \]
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Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 2715, 8, 2713} \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 a^2 d}-\frac {11 \sin ^3(c+d x) \cos (c+d x)}{24 a^2 d}-\frac {11 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {11 x}{16 a^2} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2836
Rule 2948
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \sin ^4(c+d x)-2 a^2 \sin ^5(c+d x)+a^2 \sin ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \sin ^4(c+d x) \, dx}{a^2}+\frac {\int \sin ^6(c+d x) \, dx}{a^2}-\frac {2 \int \sin ^5(c+d x) \, dx}{a^2} \\ & = -\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac {3 \int \sin ^2(c+d x) \, dx}{4 a^2}+\frac {5 \int \sin ^4(c+d x) \, dx}{6 a^2}+\frac {2 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = \frac {2 \cos (c+d x)}{a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac {3 \int 1 \, dx}{8 a^2}+\frac {5 \int \sin ^2(c+d x) \, dx}{8 a^2} \\ & = \frac {3 x}{8 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d}+\frac {5 \int 1 \, dx}{16 a^2} \\ & = \frac {11 x}{16 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {660 c+660 d x+1200 \cos (c+d x)-200 \cos (3 (c+d x))+24 \cos (5 (c+d x))-465 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))}{960 a^2 d} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {660 d x -200 \cos \left (3 d x +3 c \right )+1200 \cos \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )+24 \cos \left (5 d x +5 c \right )+75 \sin \left (4 d x +4 c \right )-465 \sin \left (2 d x +2 c \right )+1024}{960 d \,a^{2}}\) | \(78\) |
risch | \(\frac {11 x}{16 a^{2}}+\frac {5 \cos \left (d x +c \right )}{4 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{40 d \,a^{2}}+\frac {5 \sin \left (4 d x +4 c \right )}{64 d \,a^{2}}-\frac {5 \cos \left (3 d x +3 c \right )}{24 d \,a^{2}}-\frac {31 \sin \left (2 d x +2 c \right )}{64 d \,a^{2}}\) | \(107\) |
derivativedivides | \(\frac {\frac {32 \left (\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {187 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {47 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {47 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {187 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
default | \(\frac {\frac {32 \left (\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {187 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {47 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {47 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {187 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
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Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {96 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} + 165 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 38 \, \cos \left (d x + c\right )^{3} + 63 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{240 \, a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2271 vs. \(2 (122) = 244\).
Time = 53.17 (sec) , antiderivative size = 2271, normalized size of antiderivative = 17.60 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (117) = 234\).
Time = 0.30 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.74 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1536 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {935 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3840 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1410 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2560 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1410 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {935 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {165 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 256}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {165 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {165 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 935 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 935 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1536 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \]
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Time = 12.73 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {11\,x}{16\,a^2}+\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {187\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {187\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {32}{15}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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