Integrand size = 27, antiderivative size = 73 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d} \]
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Time = 0.05 (sec) , antiderivative size = 73, 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} \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {a^3 \csc ^3(c+d x)}{3 d} \]
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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)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^6 \text {Subst}\left (\int \frac {(a+x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \left (\frac {a^3}{x^7}+\frac {3 a^2}{x^6}+\frac {3 a}{x^5}+\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d} \]
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Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(50\) |
default | \(-\frac {a^{3} \left (\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(50\) |
parallelrisch | \(\frac {a^{3} \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (105 \cos \left (2 d x +2 c \right )-5 \cos \left (6 d x +6 c \right )-408 \sin \left (d x +c \right )+40 \sin \left (3 d x +3 c \right )+30 \cos \left (4 d x +4 c \right )-210\right )}{30720 d}\) | \(85\) |
risch | \(\frac {4 i a^{3} \left (45 i {\mathrm e}^{8 i \left (d x +c \right )}+10 \,{\mathrm e}^{9 i \left (d x +c \right )}-130 i {\mathrm e}^{6 i \left (d x +c \right )}-102 \,{\mathrm e}^{7 i \left (d x +c \right )}+45 i {\mathrm e}^{4 i \left (d x +c \right )}+102 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\) | \(104\) |
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.18 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {45 \, a^{3} \cos \left (d x + c\right )^{2} - 55 \, a^{3} + 4 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{2} - 14 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \]
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Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \]
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Time = 9.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3\,\left (-20\,{\sin \left (c+d\,x\right )}^6+20\,{\sin \left (c+d\,x\right )}^3+45\,{\sin \left (c+d\,x\right )}^2+36\,\sin \left (c+d\,x\right )+10\right )}{60\,d\,{\sin \left (c+d\,x\right )}^6} \]
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