Integrand size = 27, antiderivative size = 86 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {2 a \log (\sin (c+d x))}{d}-\frac {2 b \sin (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}+\frac {b \sin ^3(c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2916, 12, 780} \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {2 a \log (\sin (c+d x))}{d}+\frac {b \sin ^3(c+d x)}{3 d}-\frac {2 b \sin (c+d x)}{d}-\frac {b \csc (c+d x)}{d} \]
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Rule 12
Rule 780
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^3 (a+x) \left (b^2-x^2\right )^2}{x^3} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+x) \left (b^2-x^2\right )^2}{x^3} \, dx,x,b \sin (c+d x)\right )}{b^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 b^2+\frac {a b^4}{x^3}+\frac {b^4}{x^2}-\frac {2 a b^2}{x}+a x+x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d} \\ & = -\frac {b \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {2 a \log (\sin (c+d x))}{d}-\frac {2 b \sin (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}+\frac {b \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc (c+d x)}{d}-\frac {2 b \sin (c+d x)}{d}+\frac {b \sin ^3(c+d x)}{3 d}-\frac {a \left (\csc ^2(c+d x)+4 \log (\sin (c+d x))-\sin ^2(c+d x)\right )}{2 d} \]
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Time = 0.50 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(105\) |
default | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(105\) |
parallelrisch | \(\frac {128 a \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-128 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \csc \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )-\frac {\cos \left (4 d x +4 c \right )}{9}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {80 b \left (\cos \left (2 d x +2 c \right )+\frac {\cos \left (4 d x +4 c \right )}{20}-\frac {9}{4}\right )}{27}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}\) | \(122\) |
risch | \(2 i a x +\frac {i b \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {7 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {7 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i b \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {4 i a c}{d}-\frac {2 i \left (i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{3 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(183\) |
norman | \(\frac {-\frac {a}{8 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {6 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {25 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {6 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {21 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(206\) |
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Time = 0.41 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {6 \, a \cos \left (d x + c\right )^{4} - 9 \, a \cos \left (d x + c\right )^{2} + 24 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (b \cos \left (d x + c\right )^{4} + 4 \, b \cos \left (d x + c\right )^{2} - 8 \, b\right )} \sin \left (d x + c\right ) - 3 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 \, b \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left (\sin \left (d x + c\right )\right ) - 12 \, b \sin \left (d x + c\right ) - \frac {3 \, {\left (2 \, b \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 \, b \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 12 \, b \sin \left (d x + c\right ) + \frac {3 \, {\left (6 \, a \sin \left (d x + c\right )^{2} - 2 \, b \sin \left (d x + c\right ) - a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
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Time = 11.43 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.66 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {18\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {82\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+22\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {2\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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