Integrand size = 29, antiderivative size = 125 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d}-\frac {\left (2 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {2 a b \sin ^2(c+d x)}{d}+\frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {a b \sin ^4(c+d x)}{2 d}+\frac {b^2 \sin ^5(c+d x)}{5 d} \]
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Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 962} \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 d}-\frac {\left (2 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {a b \sin ^4(c+d x)}{2 d}-\frac {2 a b \sin ^2(c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d}+\frac {b^2 \sin ^5(c+d x)}{5 d} \]
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Rule 12
Rule 962
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^2 (a+x)^2 \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a^2 b^2 \left (1-\frac {b^2}{2 a^2}\right )+\frac {a^2 b^4}{x^2}+\frac {2 a b^4}{x}-4 a b^2 x+\left (a^2-2 b^2\right ) x^2+2 a x^3+x^4\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = -\frac {a^2 \csc (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d}-\frac {\left (2 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {2 a b \sin ^2(c+d x)}{d}+\frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {a b \sin ^4(c+d x)}{2 d}+\frac {b^2 \sin ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {b^2 \sin (c+d x)}{d}-\frac {2 a b \sin ^2(c+d x)}{d}+\frac {a^2 \sin ^3(c+d x)}{3 d}-\frac {2 b^2 \sin ^3(c+d x)}{3 d}+\frac {a b \sin ^4(c+d x)}{2 d}+\frac {b^2 \sin ^5(c+d x)}{5 d} \]
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Time = 0.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+2 a b \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+\frac {b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(120\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+2 a b \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+\frac {b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(120\) |
parallelrisch | \(\frac {-24 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +5 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )+\frac {\cos \left (4 d x +4 c \right )}{20}-\frac {9}{4}\right ) a^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+9 b \left (a \cos \left (2 d x +2 c \right )+\frac {\cos \left (4 d x +4 c \right ) a}{12}+\frac {b \sin \left (5 d x +5 c \right )}{60}+\frac {5 b \sin \left (d x +c \right )}{6}+\frac {5 b \sin \left (3 d x +3 c \right )}{36}-\frac {13 a}{12}\right )}{12 d}\) | \(145\) |
risch | \(-2 i x a b +\frac {3 a b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {7 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 d}-\frac {7 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 d}+\frac {3 a b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {4 i a b c}{d}-\frac {2 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\sin \left (5 d x +5 c \right ) b^{2}}{80 d}+\frac {a b \cos \left (4 d x +4 c \right )}{16 d}-\frac {a^{2} \sin \left (3 d x +3 c \right )}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) b^{2}}{48 d}\) | \(238\) |
norman | \(\frac {-\frac {a^{2}}{2 d}-\frac {a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {\left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (7 a^{2}-2 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (125 a^{2}-16 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (125 a^{2}-16 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {2 \left (215 a^{2}-58 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {8 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(302\) |
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Time = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {48 \, b^{2} \cos \left (d x + c\right )^{6} - 16 \, {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} - 480 \, a b \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 64 \, {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 640 \, a^{2} - 128 \, b^{2} - 15 \, {\left (8 \, a b \cos \left (d x + c\right )^{4} + 16 \, a b \cos \left (d x + c\right )^{2} - 11 \, a b\right )} \sin \left (d x + c\right )}{240 \, d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {6 \, b^{2} \sin \left (d x + c\right )^{5} + 15 \, a b \sin \left (d x + c\right )^{4} - 60 \, a b \sin \left (d x + c\right )^{2} + 10 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{3} + 60 \, a b \log \left (\sin \left (d x + c\right )\right ) - 30 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right ) - \frac {30 \, a^{2}}{\sin \left (d x + c\right )}}{30 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {6 \, b^{2} \sin \left (d x + c\right )^{5} + 15 \, a b \sin \left (d x + c\right )^{4} + 10 \, a^{2} \sin \left (d x + c\right )^{3} - 20 \, b^{2} \sin \left (d x + c\right )^{3} - 60 \, a b \sin \left (d x + c\right )^{2} + 60 \, a b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 60 \, a^{2} \sin \left (d x + c\right ) + 30 \, b^{2} \sin \left (d x + c\right ) - \frac {30 \, {\left (2 \, a b \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )}}{30 \, d} \]
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Time = 11.99 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.56 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {16\,a\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {8\,a\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}-\frac {16\,a\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d}+\frac {8\,a\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}+\frac {20\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {16\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {8\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {22\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {256\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {368\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {96\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {32\,b^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {2\,a\,b\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {2\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {9\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {2\,b^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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