Integrand size = 29, antiderivative size = 132 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {7 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3855, 2687, 30, 3853} \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {7 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2952
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cot ^4(c+d x) \csc (c+d x)+2 a^2 \cot ^4(c+d x) \csc ^2(c+d x)+a^2 \cot ^4(c+d x) \csc ^3(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^4(c+d x) \csc (c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} a^2 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{4} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} a^2 \int \csc ^3(c+d x) \, dx+\frac {1}{8} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = -\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} a^2 \int \csc (c+d x) \, dx \\ & = -\frac {7 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(132)=264\).
Time = 0.27 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.02 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=a^2 \left (-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{5 d}+\frac {9 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {7 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{80 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{80 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {7 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {7 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {9 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{5 d}-\frac {7 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{80 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{80 d}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.30
method | result | size |
parallelrisch | \(-\frac {\left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {24 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-51 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+51 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-168 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{384 d}\) | \(172\) |
risch | \(-\frac {a^{2} \left (135 \,{\mathrm e}^{11 i \left (d x +c \right )}-445 \,{\mathrm e}^{9 i \left (d x +c \right )}+480 i {\mathrm e}^{10 i \left (d x +c \right )}-330 \,{\mathrm e}^{7 i \left (d x +c \right )}-480 i {\mathrm e}^{8 i \left (d x +c \right )}-330 \,{\mathrm e}^{5 i \left (d x +c \right )}+960 i {\mathrm e}^{6 i \left (d x +c \right )}-445 \,{\mathrm e}^{3 i \left (d x +c \right )}-960 i {\mathrm e}^{4 i \left (d x +c \right )}+135 \,{\mathrm e}^{i \left (d x +c \right )}+96 i {\mathrm e}^{2 i \left (d x +c \right )}-96 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(192\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {2 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(199\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {2 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(199\) |
norman | \(\frac {-\frac {a^{2}}{384 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}-\frac {5 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {3 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {11 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {11 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {3 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {5 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {33 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {33 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {7 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(339\) |
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Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.60 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {192 \, a^{2} \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 270 \, a^{2} \cos \left (d x + c\right )^{5} + 560 \, a^{2} \cos \left (d x + c\right )^{3} - 210 \, a^{2} \cos \left (d x + c\right ) - 105 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.37 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 30 \, a^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {192 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.73 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 840 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2058 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
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Time = 11.22 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.57 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\left (5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-255\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+255\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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