Integrand size = 27, antiderivative size = 55 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^4(c+d x)}{4 d} \]
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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} \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^2(c+d x)}{2 d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^5 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^4 \text {Subst}\left (\int \frac {(a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^4(c+d x)}{4 d} \]
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Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(40\) |
default | \(-\frac {a^{2} \left (\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(40\) |
parallelrisch | \(\frac {a^{2} \left (60 \cos \left (2 d x +2 c \right )-512 \sin \left (d x +c \right )+33 \cos \left (4 d x +4 c \right )-285\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288 d}\) | \(63\) |
risch | \(\frac {2 a^{2} \left (3 \,{\mathrm e}^{6 i \left (d x +c \right )}-12 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 i {\mathrm e}^{5 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i {\mathrm e}^{3 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\) | \(80\) |
norman | \(\frac {-\frac {a^{2}}{64 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d}-\frac {7 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {5 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {5 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {5 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {5 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {7 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {13 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(225\) |
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Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {6 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a^{2} \sin \left (d x + c\right ) - 9 \, a^{2}}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \]
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Time = 9.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {2\,a^2\,\sin \left (c+d\,x\right )}{3}+\frac {a^2}{4}}{d\,{\sin \left (c+d\,x\right )}^4} \]
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