Integrand size = 29, antiderivative size = 129 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {23 x}{16 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d} \]
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Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 2713, 2715, 8} \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac {23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2836
Rule 2948
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (a^3 \sin ^3(c+d x)-3 a^3 \sin ^4(c+d x)+3 a^3 \sin ^5(c+d x)-a^3 \sin ^6(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \sin ^3(c+d x) \, dx}{a^3}-\frac {\int \sin ^6(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^4(c+d x) \, dx}{a^3}+\frac {3 \int \sin ^5(c+d x) \, dx}{a^3} \\ & = \frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int \sin ^4(c+d x) \, dx}{6 a^3}-\frac {9 \int \sin ^2(c+d x) \, dx}{4 a^3}-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = -\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int \sin ^2(c+d x) \, dx}{8 a^3}-\frac {9 \int 1 \, dx}{8 a^3} \\ & = -\frac {9 x}{8 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int 1 \, dx}{16 a^3} \\ & = -\frac {23 x}{16 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(129)=258\).
Time = 1.70 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-3 (3+920 d x) \cos \left (\frac {c}{2}\right )-2520 \cos \left (\frac {c}{2}+d x\right )-2520 \cos \left (\frac {3 c}{2}+d x\right )+945 \cos \left (\frac {3 c}{2}+2 d x\right )-945 \cos \left (\frac {5 c}{2}+2 d x\right )+380 \cos \left (\frac {5 c}{2}+3 d x\right )+380 \cos \left (\frac {7 c}{2}+3 d x\right )-135 \cos \left (\frac {7 c}{2}+4 d x\right )+135 \cos \left (\frac {9 c}{2}+4 d x\right )-36 \cos \left (\frac {9 c}{2}+5 d x\right )-36 \cos \left (\frac {11 c}{2}+5 d x\right )+5 \cos \left (\frac {11 c}{2}+6 d x\right )-5 \cos \left (\frac {13 c}{2}+6 d x\right )+9 \sin \left (\frac {c}{2}\right )-2760 d x \sin \left (\frac {c}{2}\right )+2520 \sin \left (\frac {c}{2}+d x\right )-2520 \sin \left (\frac {3 c}{2}+d x\right )+945 \sin \left (\frac {3 c}{2}+2 d x\right )+945 \sin \left (\frac {5 c}{2}+2 d x\right )-380 \sin \left (\frac {5 c}{2}+3 d x\right )+380 \sin \left (\frac {7 c}{2}+3 d x\right )-135 \sin \left (\frac {7 c}{2}+4 d x\right )-135 \sin \left (\frac {9 c}{2}+4 d x\right )+36 \sin \left (\frac {9 c}{2}+5 d x\right )-36 \sin \left (\frac {11 c}{2}+5 d x\right )+5 \sin \left (\frac {11 c}{2}+6 d x\right )+5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {-1380 d x -36 \cos \left (5 d x +5 c \right )+380 \cos \left (3 d x +3 c \right )-2520 \cos \left (d x +c \right )+5 \sin \left (6 d x +6 c \right )-135 \sin \left (4 d x +4 c \right )+945 \sin \left (2 d x +2 c \right )-2176}{960 a^{3} d}\) | \(78\) |
risch | \(-\frac {23 x}{16 a^{3}}-\frac {21 \cos \left (d x +c \right )}{8 a^{3} d}+\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{3}}-\frac {3 \cos \left (5 d x +5 c \right )}{80 d \,a^{3}}-\frac {9 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}+\frac {19 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {63 \sin \left (2 d x +2 c \right )}{64 d \,a^{3}}\) | \(107\) |
derivativedivides | \(\frac {\frac {16 \left (-\frac {23 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {391 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {75 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {17 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {75 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {391 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}-\frac {17}{60}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{3} d}\) | \(168\) |
default | \(\frac {\frac {16 \left (-\frac {23 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {391 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {75 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {17 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {75 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {391 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}-\frac {17}{60}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{3} d}\) | \(168\) |
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {144 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} + 345 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 62 \, \cos \left (d x + c\right )^{3} + 123 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, a^{3} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2404 vs. \(2 (122) = 244\).
Time = 133.53 (sec) , antiderivative size = 2404, normalized size of antiderivative = 18.64 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (117) = 234\).
Time = 0.42 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.89 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3264 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {7680 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2250 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5440 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2250 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1955 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {345 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 544}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {345 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {345 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3264 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 544\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \]
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Time = 13.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {23\,x}{16\,a^3}-\frac {\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {23\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {68}{15}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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