Integrand size = 27, antiderivative size = 55 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^6(c+d x)}{6 d} \]
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Time = 0.05 (sec) , antiderivative size = 55, 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 = {2912, 12, 45} \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^6(c+d x)}{6 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^4(c+d x)}{4 d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^7 (a+x)^2}{x^7} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^6 \text {Subst}\left (\int \frac {(a+x)^2}{x^7} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \left (\frac {a^2}{x^7}+\frac {2 a}{x^6}+\frac {1}{x^5}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^6(c+d x)}{6 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^6(c+d x)}{6 d} \]
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Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(40\) |
default | \(-\frac {a^{2} \left (\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(40\) |
parallelrisch | \(\frac {a^{2} \left (435 \cos \left (2 d x +2 c \right )-35 \cos \left (6 d x +6 c \right )-3072 \sin \left (d x +c \right )+210 \cos \left (4 d x +4 c \right )-1890\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{491520 d}\) | \(74\) |
risch | \(-\frac {4 a^{2} \left (15 \,{\mathrm e}^{8 i \left (d x +c \right )}-70 \,{\mathrm e}^{6 i \left (d x +c \right )}+48 i {\mathrm e}^{7 i \left (d x +c \right )}+15 \,{\mathrm e}^{4 i \left (d x +c \right )}-48 i {\mathrm e}^{5 i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\) | \(80\) |
norman | \(\frac {-\frac {a^{2}}{384 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}-\frac {7 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {7 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {21 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {7 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {7 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {21 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {7 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {7 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {17 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(301\) |
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Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \cos \left (d x + c\right )^{2} - 24 \, a^{2} \sin \left (d x + c\right ) - 25 \, a^{2}}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} + 24 \, a^{2} \sin \left (d x + c\right ) + 10 \, a^{2}}{60 \, d \sin \left (d x + c\right )^{6}} \]
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Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} + 24 \, a^{2} \sin \left (d x + c\right ) + 10 \, a^{2}}{60 \, d \sin \left (d x + c\right )^{6}} \]
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Time = 9.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{4}+\frac {2\,a^2\,\sin \left (c+d\,x\right )}{5}+\frac {a^2}{6}}{d\,{\sin \left (c+d\,x\right )}^6} \]
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