Integrand size = 29, antiderivative size = 87 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7 x}{8 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d} \]
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Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 2715, 8, 2713} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac {7 x}{8 a^2} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2836
Rule 2948
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^2(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \sin ^2(c+d x)-2 a^2 \sin ^3(c+d x)+a^2 \sin ^4(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \sin ^2(c+d x) \, dx}{a^2}+\frac {\int \sin ^4(c+d x) \, dx}{a^2}-\frac {2 \int \sin ^3(c+d x) \, dx}{a^2} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}+\frac {\int 1 \, dx}{2 a^2}+\frac {3 \int \sin ^2(c+d x) \, dx}{4 a^2}+\frac {2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = \frac {x}{2 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}+\frac {3 \int 1 \, dx}{8 a^2} \\ & = \frac {7 x}{8 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(87)=174\).
Time = 0.91 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.97 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {168 d x \cos \left (\frac {c}{2}\right )+144 \cos \left (\frac {c}{2}+d x\right )+144 \cos \left (\frac {3 c}{2}+d x\right )-48 \cos \left (\frac {3 c}{2}+2 d x\right )+48 \cos \left (\frac {5 c}{2}+2 d x\right )-16 \cos \left (\frac {5 c}{2}+3 d x\right )-16 \cos \left (\frac {7 c}{2}+3 d x\right )+3 \cos \left (\frac {7 c}{2}+4 d x\right )-3 \cos \left (\frac {9 c}{2}+4 d x\right )+8 \sin \left (\frac {c}{2}\right )+168 d x \sin \left (\frac {c}{2}\right )-144 \sin \left (\frac {c}{2}+d x\right )+144 \sin \left (\frac {3 c}{2}+d x\right )-48 \sin \left (\frac {3 c}{2}+2 d x\right )-48 \sin \left (\frac {5 c}{2}+2 d x\right )+16 \sin \left (\frac {5 c}{2}+3 d x\right )-16 \sin \left (\frac {7 c}{2}+3 d x\right )+3 \sin \left (\frac {7 c}{2}+4 d x\right )+3 \sin \left (\frac {9 c}{2}+4 d x\right )}{192 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {84 d x -16 \cos \left (3 d x +3 c \right )+144 \cos \left (d x +c \right )+3 \sin \left (4 d x +4 c \right )-48 \sin \left (2 d x +2 c \right )+128}{96 d \,a^{2}}\) | \(56\) |
risch | \(\frac {7 x}{8 a^{2}}+\frac {3 \cos \left (d x +c \right )}{2 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{6 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(73\) |
derivativedivides | \(\frac {\frac {8 \left (\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {15 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(114\) |
default | \(\frac {\frac {8 \left (\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {15 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(114\) |
norman | \(\frac {\frac {357 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {133 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {525 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {63 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {455 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {231 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {525 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {455 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {357 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {231 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {8}{3 a d}+\frac {133 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {63 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {7 x}{8 a}+\frac {63 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {75 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {21 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {7 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {21 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {101 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {21 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {893 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {229 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {99 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {339 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {1187 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {113 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {205 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {25 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(565\) |
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Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {16 \, \cos \left (d x + c\right )^{3} - 21 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 48 \, \cos \left (d x + c\right )}{24 \, a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (82) = 164\).
Time = 20.55 (sec) , antiderivative size = 1153, normalized size of antiderivative = 13.25 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (79) = 158\).
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {128 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {21 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 32}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {21 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{12 \, d} \]
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Time = 0.70 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {21 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 128 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \]
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Time = 9.90 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7\,x}{8\,a^2}+\frac {2\,\cos \left (c+d\,x\right )}{a^2\,d}-\frac {2\,{\cos \left (c+d\,x\right )}^3}{3\,a^2\,d}+\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^2\,d}-\frac {9\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^2\,d} \]
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