Integrand size = 8, antiderivative size = 28 \[ \int \cot ^3(a+b x) \, dx=-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]
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Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \cot ^3(a+b x) \, dx=-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^2(a+b x)}{2 b}-\int \cot (a+b x) \, dx \\ & = -\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \cot ^3(a+b x) \, dx=-\frac {\cot ^2(a+b x)+2 \log (\cos (a+b x))+2 \log (\tan (a+b x))}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {\cot \left (b x +a \right )^{2}}{2}+\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2}}{b}\) | \(29\) |
default | \(\frac {-\frac {\cot \left (b x +a \right )^{2}}{2}+\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2}}{b}\) | \(29\) |
parallelrisch | \(\frac {-\cot \left (b x +a \right )^{2}-2 \ln \left (\tan \left (b x +a \right )\right )+\ln \left (\sec \left (b x +a \right )^{2}\right )}{2 b}\) | \(35\) |
norman | \(-\frac {1}{2 b \tan \left (b x +a \right )^{2}}-\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}+\frac {\ln \left (1+\tan \left (b x +a \right )^{2}\right )}{2 b}\) | \(43\) |
risch | \(i x +\frac {2 i a}{b}+\frac {2 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(57\) |
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none
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \cot ^3(a+b x) \, dx=-\frac {{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \cot ^3(a+b x) \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\x \cot ^{3}{\left (a \right )} & \text {for}\: b = 0 \\\tilde {\infty } x & \text {for}\: a = - b x \\\frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tan ^{2}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \cot ^3(a+b x) \, dx=-\frac {\frac {1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.21 \[ \int \cot ^3(a+b x) \, dx=\frac {\frac {{\left (\frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 4 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 8 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{8 \, b} \]
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Time = 15.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \cot ^3(a+b x) \, dx=x\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b}+\frac {2}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {2}{b\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]
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