Integrand size = 29, antiderivative size = 91 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{8 a}+\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos (c+d x) \sin (c+d x)}{8 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d} \]
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Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2648, 2715, 8, 2645, 14} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {\sin (c+d x) \cos (c+d x)}{8 a d}+\frac {x}{8 a} \]
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Rule 8
Rule 14
Rule 2645
Rule 2648
Rule 2715
Rule 2918
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a} \\ & = -\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \cos ^2(c+d x) \, dx}{4 a}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos (c+d x) \sin (c+d x)}{8 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int 1 \, dx}{8 a}+\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {x}{8 a}+\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos (c+d x) \sin (c+d x)}{8 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(91)=182\).
Time = 1.63 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {120 d x \cos \left (\frac {c}{2}\right )+60 \cos \left (\frac {c}{2}+d x\right )+60 \cos \left (\frac {3 c}{2}+d x\right )+10 \cos \left (\frac {5 c}{2}+3 d x\right )+10 \cos \left (\frac {7 c}{2}+3 d x\right )-15 \cos \left (\frac {7 c}{2}+4 d x\right )+15 \cos \left (\frac {9 c}{2}+4 d x\right )-6 \cos \left (\frac {9 c}{2}+5 d x\right )-6 \cos \left (\frac {11 c}{2}+5 d x\right )+120 \sin \left (\frac {c}{2}\right )+120 d x \sin \left (\frac {c}{2}\right )-60 \sin \left (\frac {c}{2}+d x\right )+60 \sin \left (\frac {3 c}{2}+d x\right )-10 \sin \left (\frac {5 c}{2}+3 d x\right )+10 \sin \left (\frac {7 c}{2}+3 d x\right )-15 \sin \left (\frac {7 c}{2}+4 d x\right )-15 \sin \left (\frac {9 c}{2}+4 d x\right )+6 \sin \left (\frac {9 c}{2}+5 d x\right )-6 \sin \left (\frac {11 c}{2}+5 d x\right )}{960 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {60 d x +10 \cos \left (3 d x +3 c \right )-6 \cos \left (5 d x +5 c \right )-15 \sin \left (4 d x +4 c \right )+60 \cos \left (d x +c \right )+64}{480 d a}\) | \(56\) |
risch | \(\frac {x}{8 a}+\frac {\cos \left (d x +c \right )}{8 a d}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}-\frac {\sin \left (4 d x +4 c \right )}{32 d a}+\frac {\cos \left (3 d x +3 c \right )}{48 a d}\) | \(73\) |
derivativedivides | \(\frac {\frac {8 \left (\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(129\) |
default | \(\frac {\frac {8 \left (\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(129\) |
norman | \(\frac {\frac {15 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {5 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {5 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {15 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {1}{60 a d}+\frac {x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {x}{8 a}-\frac {5 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{30 d a}-\frac {23 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {27 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {11 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(490\) |
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {24 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} - 15 \, d x + 15 \, {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1464 vs. \(2 (70) = 140\).
Time = 11.42 (sec) , antiderivative size = 1464, normalized size of antiderivative = 16.09 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.05 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {80 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {90 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {80 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {240 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {90 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 16}{a + \frac {5 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{60 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.40 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 90 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 80 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a}}{120 \, d} \]
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Time = 13.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{8\,a}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {4}{15}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
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