Integrand size = 21, antiderivative size = 96 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {11 \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))} \]
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Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2788, 3855, 3852, 8, 3853, 3862} \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {11 \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (\csc (c+d x)+1)} \]
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Rule 8
Rule 2788
Rule 3852
Rule 3853
Rule 3855
Rule 3862
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (4 a-4 a \csc (c+d x)+4 a \csc ^2(c+d x)-3 a \csc ^3(c+d x)+a \csc ^4(c+d x)-\frac {4 a}{1+\csc (c+d x)}\right ) \, dx}{a^4} \\ & = \frac {4 x}{a^3}+\frac {\int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{a^3}-\frac {4 \int \csc (c+d x) \, dx}{a^3}+\frac {4 \int \csc ^2(c+d x) \, dx}{a^3}-\frac {4 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^3} \\ & = \frac {4 x}{a^3}+\frac {4 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))}-\frac {3 \int \csc (c+d x) \, dx}{2 a^3}+\frac {4 \int -1 \, dx}{a^3}-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {4 \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d} \\ & = \frac {11 \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(251\) vs. \(2(96)=192\).
Time = 3.92 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.61 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^5 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (-4 \sin ^8\left (\frac {1}{2} (c+d x)\right )-8 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x) (-2+7 \sin (c+d x))+\frac {1}{4} \sin ^4(c+d x) \left (-8+\cot \left (\frac {1}{2} (c+d x)\right )+28 \sin (c+d x)\right )-\frac {1}{2} \sin ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^3(c+d x) \left (9+\left (-28+66 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-66 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )+\sin ^4\left (\frac {1}{2} (c+d x)\right ) \sin ^2(c+d x) \left (9-2 \left (62+33 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-33 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )\right )}{12 a^3 d (1+\sin (c+d x))^3} \]
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Time = 0.46 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {19}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {64}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{3}}\) | \(113\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {19}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {64}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{3}}\) | \(113\) |
parallelrisch | \(\frac {\left (-132 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-132\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+306 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(132\) |
risch | \(-\frac {33 i {\mathrm e}^{5 i \left (d x +c \right )}-96 \,{\mathrm e}^{4 i \left (d x +c \right )}+33 \,{\mathrm e}^{6 i \left (d x +c \right )}-60 i {\mathrm e}^{3 i \left (d x +c \right )}+123 \,{\mathrm e}^{2 i \left (d x +c \right )}+19 i {\mathrm e}^{i \left (d x +c \right )}-52}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d \,a^{3}}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}\) | \(148\) |
norman | \(\frac {-\frac {107 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}-\frac {11 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {11 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {301 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {263 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {863 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {2221 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(245\) |
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Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (98) = 196\).
Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.15 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {104 \, \cos \left (d x + c\right )^{4} + 38 \, \cos \left (d x + c\right )^{3} - 156 \, \cos \left (d x + c\right )^{2} + 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (52 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )^{2} - 45 \, \cos \left (d x + c\right ) - 24\right )} \sin \left (d x + c\right ) - 42 \, \cos \left (d x + c\right ) + 48}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d - {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (98) = 196\).
Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.07 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {249 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1}{\frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {57 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}} - \frac {132 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{24 \, d} \]
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Time = 0.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {132 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {192}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {242 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 57 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}}}{24 \, d} \]
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Time = 9.57 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^3\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}-\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d} \]
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