Integrand size = 25, antiderivative size = 85 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 a \csc (c+d x)}{d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {2 b \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin ^2(c+d x)}{2 d} \]
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Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2916, 12, 780} \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc (c+d x)}{d}+\frac {b \sin ^2(c+d x)}{2 d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {2 b \log (\sin (c+d x))}{d} \]
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Rule 12
Rule 780
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^4 (a+x) \left (b^2-x^2\right )^2}{x^4} \, 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^4} \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \left (a+\frac {a b^4}{x^4}+\frac {b^4}{x^3}-\frac {2 a b^2}{x^2}-\frac {2 b^2}{x}+x\right ) \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = \frac {2 a \csc (c+d x)}{d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {2 b \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {b \left (\csc ^2(c+d x)+4 \log (\sin (c+d x))-\sin ^2(c+d x)\right )}{2 d} \]
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Time = 0.55 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(121\) |
default | \(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(121\) |
parallelrisch | \(\frac {32 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -3 \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )-\frac {\cos \left (4 d x +4 c \right )}{12}-\frac {25}{36}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 b \left (\frac {1}{8}-\frac {\cos \left (4 d x +4 c \right )}{8}+\cos \left (2 d x +2 c \right )\right )}{3}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(127\) |
risch | \(2 i x b -\frac {b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {4 i b c}{d}+\frac {2 i \left (6 a \,{\mathrm e}^{5 i \left (d x +c \right )}-8 a \,{\mathrm e}^{3 i \left (d x +c \right )}-3 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a \,{\mathrm e}^{i \left (d x +c \right )}+3 i b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(177\) |
norman | \(\frac {-\frac {a}{24 d}+\frac {19 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {55 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {55 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {19 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {9 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(206\) |
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Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {12 \, a \cos \left (d x + c\right )^{4} - 48 \, a \cos \left (d x + c\right )^{2} + 24 \, {\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \, {\left (2 \, b \cos \left (d x + c\right )^{4} - 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) + 32 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 \, b \sin \left (d x + c\right )^{2} - 12 \, b \log \left (\sin \left (d x + c\right )\right ) + 6 \, a \sin \left (d x + c\right ) + \frac {12 \, a \sin \left (d x + c\right )^{2} - 3 \, b \sin \left (d x + c\right ) - 2 \, a}{\sin \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 \, b \sin \left (d x + c\right )^{2} - 12 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 6 \, a \sin \left (d x + c\right ) + \frac {22 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 3 \, b \sin \left (d x + c\right ) - 2 \, a}{\sin \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 11.43 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.56 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx=\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {89\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {7\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {2\,b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {2\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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