Integrand size = 8, antiderivative size = 57 \[ \int \cot ^8(a+b x) \, dx=x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \]
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Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8} \[ \int \cot ^8(a+b x) \, dx=-\frac {\cot ^7(a+b x)}{7 b}+\frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot (a+b x)}{b}+x \]
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Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^7(a+b x)}{7 b}-\int \cot ^6(a+b x) \, dx \\ & = \frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b}+\int \cot ^4(a+b x) \, dx \\ & = -\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b}-\int \cot ^2(a+b x) \, dx \\ & = \frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b}+\int 1 \, dx \\ & = x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot ^5(a+b x)}{5 b}-\frac {\cot ^7(a+b x)}{7 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.58 \[ \int \cot ^8(a+b x) \, dx=-\frac {\cot ^7(a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(a+b x)\right )}{7 b} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {-15 \cot \left (b x +a \right )^{7}+21 \cot \left (b x +a \right )^{5}-35 \cot \left (b x +a \right )^{3}+105 b x +105 \cot \left (b x +a \right )}{105 b}\) | \(49\) |
derivativedivides | \(\frac {-\frac {\cot \left (b x +a \right )^{7}}{7}+\frac {\cot \left (b x +a \right )^{5}}{5}-\frac {\cot \left (b x +a \right )^{3}}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(52\) |
default | \(\frac {-\frac {\cot \left (b x +a \right )^{7}}{7}+\frac {\cot \left (b x +a \right )^{5}}{5}-\frac {\cot \left (b x +a \right )^{3}}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(52\) |
norman | \(\frac {\frac {\tan \left (b x +a \right )^{6}}{b}+x \tan \left (b x +a \right )^{7}-\frac {1}{7 b}+\frac {\tan \left (b x +a \right )^{2}}{5 b}-\frac {\tan \left (b x +a \right )^{4}}{3 b}}{\tan \left (b x +a \right )^{7}}\) | \(64\) |
risch | \(x +\frac {8 i \left (105 \,{\mathrm e}^{12 i \left (b x +a \right )}-315 \,{\mathrm e}^{10 i \left (b x +a \right )}+770 \,{\mathrm e}^{8 i \left (b x +a \right )}-770 \,{\mathrm e}^{6 i \left (b x +a \right )}+609 \,{\mathrm e}^{4 i \left (b x +a \right )}-203 \,{\mathrm e}^{2 i \left (b x +a \right )}+44\right )}{105 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{7}}\) | \(90\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (51) = 102\).
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.95 \[ \int \cot ^8(a+b x) \, dx=\frac {176 \, \cos \left (2 \, b x + 2 \, a\right )^{4} - 108 \, \cos \left (2 \, b x + 2 \, a\right )^{3} + 20 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 105 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b x \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 228 \, \cos \left (2 \, b x + 2 \, a\right ) - 76}{105 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \sin \left (2 \, b x + 2 \, a\right )} \]
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Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \cot ^8(a+b x) \, dx=\begin {cases} x - \frac {\cot ^{7}{\left (a + b x \right )}}{7 b} + \frac {\cot ^{5}{\left (a + b x \right )}}{5 b} - \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} + \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{8}{\left (a \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \cot ^8(a+b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {105 \, \tan \left (b x + a\right )^{6} - 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} - 15}{\tan \left (b x + a\right )^{7}}}{105 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.04 \[ \int \cot ^8(a+b x) \, dx=\frac {15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{7} - 189 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{5} + 1295 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + 13440 \, b x + 13440 \, a + \frac {9765 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{6} - 1295 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 15}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{7}} - 9765 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{13440 \, b} \]
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Time = 11.80 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75 \[ \int \cot ^8(a+b x) \, dx=x+\frac {-\frac {{\mathrm {cot}\left (a+b\,x\right )}^7}{7}+\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}+\mathrm {cot}\left (a+b\,x\right )}{b} \]
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