Integrand size = 29, antiderivative size = 91 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {3 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))} \]
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Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2953, 3045, 3855, 3852, 8, 3853, 2727} \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d} \]
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Rule 8
Rule 2727
Rule 2953
Rule 3045
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\csc ^4(c+d x) (a-a \sin (c+d x))}{a+a \sin (c+d x)} \, dx}{a^2} \\ & = \frac {\int \left (-2 \csc (c+d x)+2 \csc ^2(c+d x)-2 \csc ^3(c+d x)+\csc ^4(c+d x)+\frac {2}{1+\sin (c+d x)}\right ) \, dx}{a^2} \\ & = \frac {\int \csc ^4(c+d x) \, dx}{a^2}-\frac {2 \int \csc (c+d x) \, dx}{a^2}+\frac {2 \int \csc ^2(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \, dx}{a^2}+\frac {2 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^2} \\ & = \frac {2 \text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))}-\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac {2 \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {3 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(472\) vs. \(2(91)=182\).
Time = 1.65 (sec) , antiderivative size = 472, normalized size of antiderivative = 5.19 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-10 \cos \left (\frac {5}{2} (c+d x)\right )+20 \cos \left (\frac {7}{2} (c+d x)\right )-9 \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+9 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \cos \left (\frac {1}{2} (c+d x)\right ) \left (8+9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-3 \cos \left (\frac {3}{2} (c+d x)\right ) \left (14+9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+9 \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 \sin \left (\frac {1}{2} (c+d x)\right )+27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {1}{2} (c+d x)\right )-27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {1}{2} (c+d x)\right )-6 \sin \left (\frac {3}{2} (c+d x)\right )+27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {3}{2} (c+d x)\right )-27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {3}{2} (c+d x)\right )-2 \sin \left (\frac {5}{2} (c+d x)\right )-9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {5}{2} (c+d x)\right )+9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {5}{2} (c+d x)\right )+8 \sin \left (\frac {7}{2} (c+d x)\right )-9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )+9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )\right )}{192 a^2 d (1+\sin (c+d x))^2} \]
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Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{2}}\) | \(113\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{2}}\) | \(113\) |
parallelrisch | \(\frac {\left (-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-72\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+162 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(132\) |
risch | \(-\frac {2 \left (9 i {\mathrm e}^{5 i \left (d x +c \right )}-24 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{6 i \left (d x +c \right )}-18 i {\mathrm e}^{3 i \left (d x +c \right )}+33 \,{\mathrm e}^{2 i \left (d x +c \right )}+5 i {\mathrm e}^{i \left (d x +c \right )}-14\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}\) | \(148\) |
norman | \(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {65 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {83 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {47 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(207\) |
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Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (89) = 178\).
Time = 0.27 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.32 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {28 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} - 42 \, \cos \left (d x + c\right )^{2} + 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (14 \, \cos \left (d x + c\right )^{3} + 9 \, \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right ) - 6\right )} \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right ) + 12}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (89) = 178\).
Time = 0.20 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {27 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {129 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1}{\frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{24 \, d} \]
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Time = 0.47 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {96}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {132 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 33 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{24 \, d} \]
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Time = 9.62 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^2\,d} \]
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