Integrand size = 27, antiderivative size = 147 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\left (2 a^2-3 b^2\right ) \csc (c+d x)}{a^4 d}+\frac {b \csc ^2(c+d x)}{a^3 d}-\frac {\csc ^3(c+d x)}{3 a^2 d}+\frac {4 b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac {4 b \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a^5 d}-\frac {\left (a^2-b^2\right )^2}{a^4 b d (a+b \sin (c+d x))} \]
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Time = 0.16 (sec) , antiderivative size = 147, 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, 908} \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {b \csc ^2(c+d x)}{a^3 d}-\frac {\csc ^3(c+d x)}{3 a^2 d}+\frac {4 b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac {4 b \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a^5 d}-\frac {\left (a^2-b^2\right )^2}{a^4 b d (a+b \sin (c+d x))}+\frac {\left (2 a^2-3 b^2\right ) \csc (c+d x)}{a^4 d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^4 \left (b^2-x^2\right )^2}{x^4 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^4 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^4}{a^2 x^4}-\frac {2 b^4}{a^3 x^3}+\frac {-2 a^2 b^2+3 b^4}{a^4 x^2}+\frac {4 b^2 \left (a^2-b^2\right )}{a^5 x}+\frac {\left (a^2-b^2\right )^2}{a^4 (a+x)^2}+\frac {4 b^2 \left (-a^2+b^2\right )}{a^5 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = \frac {\left (2 a^2-3 b^2\right ) \csc (c+d x)}{a^4 d}+\frac {b \csc ^2(c+d x)}{a^3 d}-\frac {\csc ^3(c+d x)}{3 a^2 d}+\frac {4 b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac {4 b \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a^5 d}-\frac {\left (a^2-b^2\right )^2}{a^4 b d (a+b \sin (c+d x))} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {3 a \left (2 a^2-3 b^2\right ) \csc (c+d x)+3 a^2 b \csc ^2(c+d x)-a^3 \csc ^3(c+d x)+12 (a-b) b (a+b) \log (\sin (c+d x))-12 (a-b) b (a+b) \log (a+b \sin (c+d x))-\frac {3 a \left (a^2-b^2\right )^2}{b (a+b \sin (c+d x))}}{3 a^5 d} \]
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Time = 0.88 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 a^{2} \sin \left (d x +c \right )^{3}}-\frac {-2 a^{2}+3 b^{2}}{a^{4} \sin \left (d x +c \right )}+\frac {b}{a^{3} \sin \left (d x +c \right )^{2}}+\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{5}}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{a^{4} b \left (a +b \sin \left (d x +c \right )\right )}-\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(139\) |
default | \(\frac {-\frac {1}{3 a^{2} \sin \left (d x +c \right )^{3}}-\frac {-2 a^{2}+3 b^{2}}{a^{4} \sin \left (d x +c \right )}+\frac {b}{a^{3} \sin \left (d x +c \right )^{2}}+\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{5}}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{a^{4} b \left (a +b \sin \left (d x +c \right )\right )}-\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(139\) |
parallelrisch | \(\frac {-192 b \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+192 b \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \left (\cos \left (2 d x +2 c \right )-\frac {\cos \left (4 d x +4 c \right )}{12}-\frac {25}{36}\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \,a^{3} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-3\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+54 b^{2} \left (\cos \left (2 d x +2 c \right )-\frac {17}{9}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+24 a^{3} b \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+192 \sin \left (d x +c \right ) b^{4}}{48 a^{5} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(245\) |
norman | \(\frac {-\frac {1}{24 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a^{2} d}+\frac {b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a^{2} d}+\frac {\left (19 a^{2}-24 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{3} d}+\frac {\left (19 a^{2}-24 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{3} d}+\frac {\left (55 a^{4}-148 a^{2} b^{2}+96 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{5} d}+\frac {\left (55 a^{4}-148 a^{2} b^{2}+96 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{5} d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}+\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5} d}-\frac {4 b \left (a^{2}-b^{2}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{5} d}\) | \(354\) |
risch | \(-\frac {2 i \left (-12 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+16 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}-12 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-24 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+44 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-36 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+12 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-44 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+36 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-3 a^{4} {\mathrm e}^{i \left (d x +c \right )}+12 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}-12 b^{4} {\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} b \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4} d}+\frac {4 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{5} d}-\frac {4 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{5} d}\) | \(454\) |
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (145) = 290\).
Time = 0.36 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.73 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {5 \, a^{4} b - 6 \, a^{2} b^{3} - 6 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} - 12 \, {\left (a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b^{2} - a b^{4} - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 12 \, {\left (a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b^{2} - a b^{4} - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (3 \, a^{5} - 14 \, a^{3} b^{2} + 12 \, a b^{4} - 3 \, {\left (a^{5} - 4 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{5} b^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{5} b^{2} d \cos \left (d x + c\right )^{2} + a^{5} b^{2} d - {\left (a^{6} b d \cos \left (d x + c\right )^{2} - a^{6} b d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.07 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \, a^{2} b^{2} \sin \left (d x + c\right ) - a^{3} b - 3 \, {\left (a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{4}\right )} \sin \left (d x + c\right )^{3} + 6 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{2}}{a^{4} b^{2} \sin \left (d x + c\right )^{4} + a^{5} b \sin \left (d x + c\right )^{3}} - \frac {12 \, {\left (a^{2} b - b^{3}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{5}} + \frac {12 \, {\left (a^{2} b - b^{3}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.44 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {12 \, {\left (a^{2} b - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac {12 \, {\left (a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{5} b} + \frac {3 \, {\left (4 \, a^{2} b^{3} \sin \left (d x + c\right ) - 4 \, b^{5} \sin \left (d x + c\right ) - a^{5} + 6 \, a^{3} b^{2} - 5 \, a b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{5} b} - \frac {22 \, a^{2} b \sin \left (d x + c\right )^{3} - 22 \, b^{3} \sin \left (d x + c\right )^{3} - 6 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} - 3 \, a^{2} b \sin \left (d x + c\right ) + a^{3}}{a^{5} \sin \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 13.21 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.17 \[ \int \frac {\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (16\,a^2\,b-24\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (8\,a\,b^2-\frac {20\,a^3}{3}\right )-\frac {a^3}{3}+\frac {4\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (23\,a^4-44\,a^2\,b^2+16\,b^4\right )}{a}}{d\,\left (8\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,b\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {\frac {a^2}{4}+\frac {b^2}{2}}{a^4}+\frac {5}{8\,a^2}-\frac {2\,b^2}{a^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a^2\,b-4\,b^3\right )}{a^5\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^3\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (4\,a^2\,b-4\,b^3\right )}{a^5\,d} \]
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